Inconsistency of Measure-Theoretic Probability and Random Behavior of Microscopic Systems

TL;DR

This paper identifies a fundamental inconsistency in measure-theoretic probability, especially regarding infinite additivity and the interpretation of mathematical entities, which may affect understanding of microscopic systems.

Contribution

It reveals an inherent inconsistency in measure-theoretic probability and suggests resolving it through constructive mathematics to better align theory with physical reality.

Findings

Inconsistencies in probability equalities for real-valued random variables.

Contradictions arising from extended random variables and infinite additivity.

Implication that mathematical notions of numbers may cause physical misinterpretations.

Abstract

We report an inconsistency found in probability theory (also referred to as measure-theoretic probability). For probability measures induced by real-valued random variables, we deduce an "equality" such that one side of the "equality" is a probability, but the other side is not. For probability measures induced by extended random variables, we deduce an "equality" such that its two sides are unequal probabilities. The deduced expressions are erroneous only when it can be proved that measure-theoretic probability is a theory free from contradiction. However, such a proof does not exist. The inconsistency appears only in the theory rather than in the physical world, and will not affect practical applications as long as ideal events in the theory (which will not occur physically) are not mistaken for observable events in the real world. Nevertheless, unlike known paradoxes in mathematics, the inconsistency cannot be explained away and hence must be resolved. The assumption of infinite additivity in the theory is relevant to the inconsistency, and may cause confusion of ideal events and real events. As illustrated by an example in this article, since abstract properties of mathematical entities in theoretical thinking are not necessarily properties of physical quantities observed in the real world, mistaking the former for the latter may lead to misinterpreting random phenomena observed in experiments with microscopic systems. Actually the inconsistency is due to the notion of "numbers" adopted in conventional mathematics. A possible way to resolve the inconsistency is to treat "numbers" from the viewpoint of constructive mathematics.

Equations

$${\int }_{\mathbb{R}}fd{\rho }_n\to {\int }_{\mathbb{R}}fd\rho$$

$$E_2=H(E_1),E_3=H(E_2)=H(H(E_1)=H^{(2)}(E_1),\cdots ,E_n=H^{(n-1)}(E_1)$$

$${\delta }_n(\{n\})=\mathbb{P}(W_n=n)=1$$

$$\underset{n\to \infty }{\lim }[{\delta }_n(\{n\})]=1.$$

$$E_n=H(\{n-1\})=\{n\}$$

$$[\underset{n\to \infty }{\lim }{\rho }_n](E)=\underset{n\to \infty }{\lim }[{\rho }_n(E_n)]$$

$${\delta }_{1/n}(E_n)={\delta }_{1/n}(E_1)=\mathbb{P}(Q_n\in E_1)=0.$$

$$\underset{n\to \infty }{\lim }[{\delta }_{1/n}(E_1)]=0.$$

$${\rho }_n(E_n)={\delta }_{1/n}(\{n\})=\mathbb{P}(Q_n=n)=0.$$

$$\underset{n\to \infty }{\lim }[{\rho }_n(E_n)]=0$$

$$|{\rho }_n(E_n)-{\rho }_n^{\prime }(E_n^{\prime })|\to 0,$$

$${\mu_{n}(\{j\})={\mathbb{P}}(Z_{n}=j)=}\left\{\begin{array}[]{ll}{(1-q)q^{n-j},}&\mbox{{$j=1,2,\cdots,n-1$}}\\ {1-q+q^{n},}&\mbox{{$j=n$}}\end{array}\right.$$

$$(1-q)+(1-q)\sum _{k\ge 1}{q}^k=1$$

$$\mathbb{P}(Z_n=n)={\mu }_n(\{n\})=1-q+q^n.$$

$$\underset{n\to \infty }{\lim }[{\mu }_n(\{n\})]=\underset{n\to \infty }{\lim }(1-q+q^n)=1-q.$$

$$\mathbb{P}(Z_n<n)={\mu }_n(E_n^{\prime })=1-\mathbb{P}(Z_n=n)=1-{\mu }_n(\{n\})=q-q^n$$

$$E_n^{\prime }=H^{\prime }(E_{n-1}^{\prime })=E_{n-1}^{\prime }\cup [n-1,n)=(-\infty ,n),n>1$$

$$E^{\prime }=\underset{n\to \infty }{\lim }{H}^{\prime (n)}(E_1^{\prime })=\mathbb{R}\in \mathcal{R}.$$

$$\underset{n\to \infty }{\lim }[{\mu }_n(E_n^{\prime })]=\underset{n\to \infty }{\lim }(q-q^n)=q$$

$${\lambda_{n}(\{a\})={\mathbb{P}}(X_{n}=a)=}\left\{\begin{array}[]{ll}{q,}&\mbox{{$a=0$}}\\ {1-q,}&\mbox{{$a=1$}}\end{array}\right.$$

$$Y_n=\max \{X_j:j\in \{1,2,\cdots ,n\}\},n=1,2,\cdots .$$

$${\gamma_{n}(\{a\})={\mathbb{P}}(Y_{n}=a)=}\left\{\begin{array}[]{ll}{q^{n},}&\mbox{{$a=0$}}\\ {1-q^{n},}&\mbox{{$a=1$.}}\end{array}\right.$$

$${\gamma(\{a\})=}\left\{\begin{array}[]{ll}{0,}&\mbox{{$a=0$}}\\ {1,}&\mbox{{$a=1$}}\end{array}\right.$$

$$Z_n=\max \{j:X_j=Y_n,j\in \{1,2,\cdots ,n\}\},n=1,2,\cdots .$$

$$\{Z_n=j\}=\{X_j=1\}\cap \{X_i=0,j<i\le n\}$$

$$\{Z_n=n\}=\{X_n=1\}\cup \{X_i=0,i=1,2,\cdots ,n\}=\{X_n=1\}\cup \{Y_n=0\}.$$

$$\lambda (\{0\})=\underset{n\to \infty }{\lim }[{\mu }_n(E_n^{\prime })]$$

$$\mathbb{P}(\{X_n=0\}\setminus \{Y_n=0\})=\mathbb{P}(X_n=0)-\mathbb{P}(Y_n=0)=\mathbb{P}(Z_n<n).$$

$${\lambda }_n(\{0\})-{\gamma }_n(\{0\})={\mu }_n(E_n^{\prime }),n=1,2,\cdots .$$

$$|{\lambda }_n(\{0\})-{\mu }_n(E_n^{\prime })|={\gamma }_n(\{0\})\to 0.$$

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1 Introduction

A basic requirement in mathematics and its various applications is that the two sides of an equality must be entities of the same kind [2]. Actually, all equalities proved in a consistent (i.e., contradiction-free) theory satisfy the above requirement. In a mathematical theory, if an “equality” is deducible such that its two sides are entities of different kinds, the theory is inconsistent. No one will consider quantities of different kinds (e.g., length and area) equal, even if their numerical values happen to be identical. It is absurd even to ask whether quantities different in kind are equal, as they are calculated according to different rules.

Physical quantities (i.e., quantities in the physical world) are usually modeled by quantities in mathematics. For instance, many physical quantities are modeled by real-valued variables, functions or parameters. Unlike physical quantities modeled by their mathematical counterparts, real numbers themselves are pure numbers, which serve as one of the basic kinds of quantities in mathematics. Mathematical quantities of different kinds follow different rules in calculation. Operations for pure numbers are determined by the corresponding axioms (e.g., see [13]). In mathematics, however, not all real-valued quantities are calculated according to the same rules. For example, probabilities are values assigned by a probability measure to “events” modeled by measurable sets in a $\sigma$-algebra. One cannot calculate probabilities in the same way as one calculates values of functions which have nothing to do with “events”, even if such functions and probability measures have the same range. Similarly, probabilities and pure numbers in the closed interval $[0,1]$ are also calculated according to different rules, and hence belong to different kinds of quantities. A probability describes quantitatively how likely an event will occur. No pure number conveys such information. Rules of calculation with pure numbers in the interval $[0,1]$ are not necessarily subject to the constraints imposed on calculating probabilities.

As shown above, if two mathematical quantities both take real numbers as their values but are calculated according to different rules, they are different in kind. Let $\Gamma$ and $\Psi$ denote two real-valued expressions, which may be the limits of sequences $(\Gamma_{n})_{n\geq 1}$ and $(\Psi_{n})_{n\geq 1}$, respectively, where $\Gamma_{n}=\Psi_{n}$ for each $n$, and $\Gamma_{n}$ and $\Psi_{n}$ represent quantities of the same kind; but the quantity represented by $\Gamma$ (or $\Psi$) may or may not belong to those kinds of quantities represented by $\Gamma_{n}$ (or $\Psi_{n}$). For $\Gamma=\Psi$, the basic requirement mentioned at the beginning of this section reduces to a trite, necessary condition. We refer to this condition as “presupposition of equality” and state it below.

Presupposition of equality**.**

In a consistent mathematical theory, the two sides of $\Gamma=\Psi$ represent quantities of the same kind which are calculated based on the same rules. The rules are determined by the kind of quantities to which $\Gamma$ and $\Psi$ belong.

The measure-theoretic foundations for probability theory were laid by Kolmogorov in 1933 [8]. Since then, it has been postulated (somewhat implicitly) that measure-theoretic probability is a consistent theory. However, the postulated consistency of the theory appears to be incompatible with the presupposition of equality. In this article, we report an inconsistency of measure-theoretic probability. For probability measures induced by real-valued random variables, we deduce an “equality” such that one side of the “equality” is a probability, but the other side is not. For probability measures induced by extended random variables, we deduce an “equality” such that its two sides are unequal probabilities. Known paradoxes in mathematics are either considered issues of formal logic (such as Russell’s paradox), or treated as deducible statements which are counterintuitive but true (e.g., see [6, 14]), where “true” implies “without contradiction”. Regarding deducible statements as results without contradiction is not grounded on mathematical justification; it originates from a popular belief, i.e., mathematics is free from contradiction as it is. The belief is an issue of mathematical philosophy (e.g., see [3]), which is not our concern. Unlike the paradoxes treated in [6, 14] or discussed by logicians, the inconsistency found in measure-theoretic probability cannot be explained away.

In the rest of this article, we first deliberate why the limit of a convergent sequence from the interval $[0,1]$ may not necessarily be a limit probability, even if the terms of the sequence are all probabilities. After showing how to decide whether a limit probability can be calculated as the limit of a sequence of probabilities (Section 2), we review briefly the notion of tightness of probability measures induced by random variables (Section 3). Tightness is a condition to prevent mass from “escaping to infinity” [4, 11]. With a tight sequence of measures, we construct a sequence of measures which is not tight, and then reveal the inconsistency of measure-theoretic probability (Section 4). The inconsistency is relevant to infinite additivity, which is one of the basic assumptions in probability theory [8]. Next, we clarify an ambiguity caused by the assumption of infinite additivity (Section 5). Because of this assumption, the meaning of “a realizable event” is ambiguous. Clearly, abstract properties of mathematical entities in theoretical thinking are not necessarily properties of physical quantities observed in the real world. However, misled by the assumption of infinite additivity, some people might mistake the former for the latter; as a result, they might give problematic or even incorrect interpretations of random phenomena observed in experiments with microscopic systems. We discuss this issue in the context of an experiment in quantum physics (Section 6). Finally, we explain why measure-theoretic probability can be so successful even though it is not a consistent theory, and discuss briefly how to avoid undesirable consequences caused by the inconsistency (Section 7). Actually the inconsistency is due to the notion of “numbers” adopted in conventional mathematics. A possible way to resolve the inconsistency is to treat “numbers” from the point of view of constructive mathematics (e.g., see [5, 12]).

2 Limit Probability Measure and Limit Probability

In measure-theoretic probability, random experiments are described by probability measures together with the corresponding measurable spaces. However, when concerned with values and probability distributions of random variables, we can make no mention of the underlying measurable spaces associated with experiments. Let ${\mathbb{P}}$ be a probability measure on a measurable space $(\Omega,{\mathcal{A}})$. Denote by ${\mathcal{R}}$ the $\sigma$-algebra of Borel subsets of the real line ${\mathbb{R}}$ which is equipped with the topology induced by the usual metric. Let ${\mathscr{F}}$ be a family of probability measures induced by random variables defined on $\Omega$ measurable ${\mathcal{A}}/{\mathcal{R}}$. As the probability distributions of the random variables on $\Omega$, members of ${\mathscr{F}}$ may simply be regarded as measures defined on the measurable space $({\mathbb{R}},{\mathcal{R}})$. We do not need to specify ${\mathscr{F}}$ or $\Omega$. It suffices to know that random variables with their measures belonging to ${\mathscr{F}}$ are defined on a common sample space $\Omega$. Not all measures on $({\mathbb{R}},{\mathcal{R}})$ are members of ${\mathscr{F}}$, however. Apparently, there are measures on $({\mathbb{R}},{\mathcal{R}})$ induced by irrelevant random variables which are not defined on the sample space $\Omega$.

Let $(\rho_{n})_{n\geq 1}$ be a sequence of probability measures induced by real-valued random variables. If $(\rho_{n})_{n\geq 1}$ converges weakly, denote by $\rho$ the weak limit of $(\rho_{n})_{n\geq 1}$ where $\rho$ is a probability measure on ${\mathcal{R}}$. Such convergence means that, as $n\to\infty$

[TABLE]

for every bounded, continuous function $f$ on ${\mathbb{R}}$ [4, 11]. For probability measures $\rho,\rho_{1},\rho_{2},\cdots$ induced respectively by random variables $V,V_{1},V_{2},\cdots$, if $\rho_{n}$ converges weakly to $\rho$, $V_{n}$ is said to converge in distribution to $V$. If random variables $U$ and $V$ have the same distribution, $V_{n}$ also converges in distribution to $U$ [11]. The random variables $V,V_{1},V_{2},\cdots$ and $U$ may be defined on distinct probability spaces [4].

If a sequence $(\rho_{n})_{n\geq 1}$ converges weakly to $\rho$, and if not only the terms $\rho_{n}$ but also the weak limit $\rho$ belong to ${\mathscr{F}}$, we call $\rho$ the limit probability measure of $(\rho_{n})_{n\geq 1}$. For a limit probability measure $\rho$ and an event (measurable set) $E\in{\mathcal{R}}$, we call $\rho(E)$ a limit probability. For every $\alpha\in{\mathscr{F}}$, there is a sequence $(\alpha_{n})_{n\geq 1}$ of measures from ${\mathscr{F}}$ such that $\alpha_{n}=\alpha$ for all $n$. Hence $\alpha$ is the limit probability measure of $(\alpha_{n})_{n\geq 1}$. So a probability $\alpha(E)$ is trivially a limit probability. The presupposition of equality implies that, if one side of an equality represents a probability, so does the other side, and operations performed on the two sides must obey the same rules for calculating probabilities. If $\rho$ is the limit probability measure of $(\rho_{n})_{n\geq 1}$, we also write $\rho=[\lim_{n}\rho_{n}]$, where the brackets cannot be omitted. Accordingly, $\rho(E)=[\lim_{n}\rho_{n}](E)$. With $[\lim_{n}\rho_{n}]$ as a synonym for the limit probability measure $\rho$, we want to emphasize that rules for calculating limit probabilities and rules for calculating limits of real sequences are different. Usually, a convergent real sequence from the interval $[0,1]$ may not necessarily converge to a probability, even if the terms of the sequence are all probabilities. But there is an exception: the limit of a real sequence may “coincide” with a limit probability. We shall explain what this means shortly.

Whenever we say “events $E_{n}\in{\mathcal{R}}$ form a sequence $(E_{n})_{n\geq 1}$”, we assume that, for a given $E_{1}\in{\mathcal{R}}$, there is a map $H:{\mathcal{R}}\mapsto{\mathcal{R}}$ which defines $E_{n},n>1$ recursively as follows:

[TABLE]

and as $n\to\infty$, $E_{n}$ tends to a limit event $E=\lim_{n}H^{(n)}(E_{1})$. An example of such $H$ is the identity map, which maps any given $E_{1}\in{\mathcal{R}}$ to $E_{1}$ itself. If $H$ is not the identity map, the limit event $E$ may or may not be a set in ${\mathcal{R}}$.

We are interested in measures $\rho_{n}\in{\mathscr{F}},n=1,2,\cdots$, such that for events $E_{n}$ in $(E_{n})_{n\geq 1}$, the probabilities $\rho_{n}(E_{n})$ form a convergent sequence. We denote the limit of $(\rho_{n}(E_{n}))_{n\geq 1}$ by $\lim_{n}[\rho_{n}(E_{n})]$ rather than $\lim_{n}\rho_{n}(E_{n})$. The meaning of $\lim_{n}\rho_{n}(E_{n})$ is ambiguous. For instance, suppose that $E_{n}$ tends to a limit event $E$ and $\rho_{n}$ converges weakly to $\rho$ in a generalized sense as explained in Remark 2 (see Section 3). Then $\lim_{n}\rho_{n}(E_{n})$ may mean either $\rho(E)$ or $\lim_{n}[\rho_{n}(E_{n})]$. The former is always a probability; but the latter may or may not be a probability. The use of symbols “$\lim_{n}[\rho_{n}(E_{n})]$” and “$[\lim_{n}\rho_{n}](E)$” may help one avoid mixing up rules for calculating limits of real sequences with rules for calculating limit probabilities. When $\lim_{n}[\rho_{n}(E_{n})]$ is not a probability, it is still the limit of a real sequence.

Example 1**.**

For $n=1,2,\cdots$, let $\delta_{n}$ be the Dirac measure concentrated on $E_{n}=\{n\}$ and let $\rho_{n}=\delta_{n}$, where $\rho_{n}$ is the probability measure induced by a degenerate random variable $W_{n}=n$. Since

[TABLE]

for each $n$, one may treat $\delta_{n}(\{n\}),n=1,2,\cdots$ as a sequence of 1’s. Such treatment implies

[TABLE]

By definition, $\lim_{n}[\delta_{n}(\{n\})]$ is the trivial limit of a constant sequence. Since $\delta_{n}$ does not converge weakly to a limit defined on ${\mathcal{R}}$, $\lim_{n}[\delta_{n}(\{n\})]$ is not a limit probability. The events $E_{n},n>1$ associated with $\rho_{n}(E_{n})$ are defined by $H:{\mathcal{R}}\mapsto{\mathcal{R}}$ such that

[TABLE]

with $E_{1}=\{1\}$. Evidently, $(E_{n})_{n\geq 1}$ has no limit in ${\mathcal{R}}$ for $\lim_{n}H^{(n)}(\{1\})=\{\infty\}\not\in{\mathcal{R}}$. Colloquially, one may also express this fact by saying that all the mass “escapes to infinity” as $n\to\infty$. $\Box$

When the limit event $E$ is in ${\mathcal{R}}$ and the limit probability measure $[\lim_{n}\rho_{n}]$ exists where $[\lim_{n}\rho_{n}]\in{\mathscr{F}}$ by definition, we say that $[\lim_{n}\rho_{n}](E)$ “coincides” with $\lim_{n}[\rho_{n}(E_{n})]$, if they not only have the same numerical value but also satisfy the presupposition of equality, which requires $\lim_{n}[\rho_{n}(E_{n})]$ to be a probability. In other words, by “$[\lim_{n}\rho_{n}](E)$ and $\lim_{n}[\rho_{n}(E_{n})]$ coincide” we mean that not only

[TABLE]

holds but also both sides of (1) are probabilities.

Remark 1**.**

Suppose that, as $n\to\infty$, three sequences different in kind, i.e., a sequence of measures $\rho_{n}\in{\mathscr{F}}$, a sequence of events $E_{n}\in{\mathcal{R}}$, and a sequence of probabilities $\rho_{n}(E_{n})$ approach their respective limits: $[\lim_{n}\rho_{n}]$, a limit probability measure; $E$, a limit event; and $\lim_{n}[\rho_{n}(E_{n})]$, the limit of a real sequence $(\rho_{n}(E_{n}))_{n\geq 1}$. As a result, corresponding to $\rho_{n}(E_{n})$, there exist a limit of a real sequence $\lim_{n}[\rho_{n}(E_{n})]$ and a limit probability $[\lim_{n}\rho_{n}](E)$. If $[\lim_{n}\rho_{n}](E)$ and $\lim_{n}[\rho_{n}(E_{n})]$ coincide, the limit probability can also be calculated as the limit of the real sequence; otherwise the limit probability must be calculated as the probability assigned by measure $[\lim_{n}\rho_{n}]$ to event $E$. $\Box$

Example 2**.**

For $n=1,2,\cdots$, write $\delta_{1/n}$ for the Dirac measure concentrated on $\{1/n\}$ and let $\rho_{n}=\delta_{1/n}$, where $\rho_{n}$ is the probability measure induced by a degenerate random variable $Q_{n}=1/n$. Let $H:{\mathcal{R}}\mapsto{\mathcal{R}}$ be the identity map which defines $E_{n}=E_{1}$ for all $n>1$ with $E_{1}=(-\infty,0]$. For each $n$,

[TABLE]

By treating $\delta_{1/n}(E_{1}),n=1,2,\cdots$ as a sequence of 0’s, one has

[TABLE]

The above trivial limit of the constant sequence is not a limit probability. Actually $[\lim_{n}\delta_{1/n}]=\delta_{0}$, where $\delta_{0}$ is the Dirac measure concentrated on $\{0\}$. So $[\lim_{n}\delta_{1/n}](E_{1})=1$, which does not coincide with $\lim_{n}[\delta_{1/n}(E_{1})]$. $\Box$

Example 3**.**

For $E_{n}$ and $\rho_{n}$ defined in Example 1 and Example 2, respectively, when $n>1$,

[TABLE]

As we already know, $\lim_{n}H^{(n)}(\{1\})=\{\infty\}\not\in{\mathcal{R}}$. Clearly

[TABLE]

has nothing to do with calculating a limit probability. $\Box$

The simple examples given above involve only measures induced by degenerate random variables. Actually, even for a convergent sequence $(\rho^{\prime}_{n}(E^{\prime}_{n}))_{n\geq 1}$ of probabilities, where $\rho^{\prime}_{n},n=1,2,\cdots$ are all induced by non-degenerate random variables, the limit of $(\rho^{\prime}_{n}(E^{\prime}_{n}))_{n\geq 1}$ may not necessarily be a probability, see Example 5 in the next section.

Let $(\rho_{n})_{n\geq 1}$ be a sequence of measures from ${\mathscr{F}}$ such that $[\lim_{n}\rho_{n}]=\rho\in{\mathscr{F}}$. In other words, $\rho$ is the limit probability measure of $(\rho_{n})_{n\geq 1}$. For a given $E_{1}\in{\mathcal{R}}$, let $E_{n},n>1$ be defined by $H:{\mathcal{R}}\mapsto{\mathcal{R}}$ recursively, such that $\lim_{n}H^{(n)}(E_{1})=E\in{\mathcal{R}}$. Similarly, let $E_{n}^{\prime},n>1$ be defined recursively by $H^{\prime}:{\mathcal{R}}\mapsto{\mathcal{R}}$ with $E^{\prime}=\lim_{n}H^{\prime(n)}(E_{1}^{\prime})\in{\mathcal{R}}$ where $E_{1}^{\prime}\in{\mathcal{R}}$ is given, and let $(\rho_{n}^{\prime})_{n\geq 1}$ be a sequence of measures from ${\mathscr{F}}$. We have the following lemma.

Lemma 1**.**

Assume that the three conditions (a), (b) and (c) below are all satisfied. (a) As $n\to\infty$,

[TABLE]

(b) $\lim_{n}[\rho_{n}(E_{n})]$ coincides with $[\lim_{n}\rho_{n}](E)$, and (c) measure-theoretic probability is a consistent theory. Then $\lim_{n}[\rho_{n}^{\prime}(E_{n}^{\prime})]$ is a probability.

Proof.

Condition (a) implies $\lim_{n}[\rho_{n}(E_{n})]=\lim_{n}[\rho_{n}^{\prime}(E_{n}^{\prime})]$, which together with condition (b) implies $\rho(E)=\lim_{n}[\rho_{n}^{\prime}(E_{n}^{\prime})]$. The result to be proved then follows from condition (c) and the presupposition of equality. ∎

The above lemma is equivalent to its contrapositive. We shall use the contrapositive of Lemma 1 to reveal the inconsistency of measure-theoretic probability.

3 Tightness of Probability Measures

The sequence of the Dirac measures $(\delta_{n})_{n\geq 1}$ induced by the degenerate random variables $W_{n}=n$ in Example 1 illustrates a simple scenario of mass “escaping to infinity”. Escape of mass also happens to sequences of probability measures induced by non-degenerate random variables.

Example 4**.**

Let $(\mu_{n})_{n\geq 1}$ be a sequence of probability measures induced by random variables $Z_{n}$. For any given $n$, the probability mass function of $Z_{n}$ is determined by $\mu_{n}$ as follows.

[TABLE]

where $0<q<1$. As $n\to\infty$, all the mass generated by (2) will “escape to infinity”. For any given $j<n$, $\lim_{n}[\mu_{n}(\{j\})]=\lim_{n}(1-q)q^{n-j}=0$. For $j=n$, $\mu_{n}(\{n\})=1-q+q^{n}$ is the mass at $n$. When $j=n-k$ where $1\leq k<n$ is fixed, $\mu_{n}(\{n-k\})=(1-q)q^{k}$ is the mass at $n-k$. Similar to the unit mass at $n$ expressed by the Dirac measure $\delta_{n}$, the non-vanishing part $1-q$ of the mass $1-q+q^{n}$ at $n$ and the mass $(1-q)q^{k}$ at $n-k$ will all “escape to infinity” as $n\to\infty$. So

[TABLE]

is the total escaped mass. $\Box$

Example 5**.**

As shown in the previous example, $(\mu_{n})_{n\geq 1}$ has no limit probability measure on ${\mathcal{R}}$ because of escape of mass. From (2) we see that, for each $n$, the probability of $\{Z_{n}=n\}$ is

[TABLE]

Since $(1-q+q^{n})_{n\geq 1}$ is a convergent sequence,

[TABLE]

However, as partial mass “escaped to infinity”, $\lim_{n}[\mu_{n}(\{n\})]$ is not a probability. Similarly, the probability of $\{Z_{n}<n\}$ is

[TABLE]

where a map $H^{\prime}:{\mathcal{R}}\mapsto{\mathcal{R}}$ defines

[TABLE]

recursively with $E_{1}^{\prime}=(-\infty,1)$. As one can readily see,

[TABLE]

But

[TABLE]

is also part of escaped mass rather than a probability. $\Box$

Let $(\rho_{n})_{n\geq 1}$ be a sequence of probability measures defined on ${\mathcal{R}}$. Tightness is a condition imposed on $(\rho_{n})_{n\geq 1}$ to prevent escape of mass. If $(\rho_{n})_{n\geq 1}$ is tight, then for each $\epsilon>0$, there is an interval $I_{\epsilon}\subset{\mathbb{R}}$, such that $\sup_{n}\rho_{n}(I_{\epsilon}^{c})<\epsilon$ where $I_{\epsilon}^{c}={\mathbb{R}}\setminus I_{\epsilon}$.

Example 6**.**

Consider probability measures $\mu_{n}$ induced by $Z_{n}$ in Example 4. For each $n$, $Z_{n+1}$ takes larger values with larger probabilities. The values of $Z_{n+1}$ include $n+1$ and all the values of $Z_{n}$. The sequence $(\mu_{n})_{n\geq 1}$ is not tight. To see this, consider $\epsilon=1-q$. By (2), for all $n$, $\mu_{n}(\{n\})=1-q+q^{-n}>1-q$. In particular, for all real numbers $b>0$ and for all $n>N_{b}$, $\mu_{n}(\{n\})>1-q$, where $N_{b}$ is the smallest integer larger than or equal to $b$. Consequently, for each interval $I_{b}=(-\infty,b)$, $\sup_{n}\mu_{n}(I_{b}^{c})>1-q$. The probability mass function (2) explains not only why $(\mu_{n})_{n\geq 1}$ fails to be tight but also where all the mass goes as shown in Example 4. $\Box$

Tightness is necessary but not sufficient for $(\rho_{n})_{n\geq 1}$ to converge weakly to a limit defined on ${\mathcal{R}}$. However, if $(\rho_{n})_{n\geq 1}$ has a weak limit on ${\mathcal{R}}$, then $(\rho_{n})_{n\geq 1}$ is tight.

Example 7**.**

Denote by $\lambda$ and $\lambda_{n}$ the probability measures induced by independent and identically distributed Bernoulli random variables $X$ and $X_{n},n=1,2,\cdots$, respectively. The above random variables are all defined on a common sample space $\Omega$. The probability mass function of $X_{n}$ is

[TABLE]

where $0<q<1$. The sequence $(\lambda_{n})_{n\geq 1}$ converges weakly to $\lambda$. Let $E_{1}=\{a\}$ and $H:{\mathcal{R}}\mapsto{\mathcal{R}}$ be the identity map which defines $E_{n}=\{a\}$ for all $n>1$. So $\lim_{n}H^{(n)}(E_{1})=\{a\}$. Apparently, $[\lim_{n}\lambda_{n}](\{a\})=\lambda(\{a\})$ which coincides with $\lim_{n}[\lambda_{n}(\{a\})]$, and $(\lambda_{n})_{n\geq 1}$ is tight. $\Box$

Example 8**.**

Let $X_{n},n=1,2,\cdots$ be the Bernoulli random variables in Example 7. Denote by $Y_{n}$ the largest among $X_{1},X_{2},\cdots,X_{n}$.

[TABLE]

Let $\gamma_{n}$ be the probability measures induced by $Y_{n}$. The probability mass function of $Y_{n}$ is

[TABLE]

The sequence $(\gamma_{n})_{n\geq 1}$ converges weakly to $\gamma$ where $\gamma$ is given by

[TABLE]

i.e., $\gamma$ is the Dirac measure $\delta_{1}$ concentrated on $\{1\}$. Evidently $(\gamma_{n})_{n\geq 1}$ is tight, and $[\lim_{n}\gamma_{n}](\{a\})=\gamma(\{a\})$ which coincides with $\lim_{n}[\gamma_{n}(\{a\})]$. $\Box$

Remark 2**.**

By replacing ${\mathscr{F}}$ with a larger family ${\overline{\mathscr{F}}}\supset{\mathscr{F}}$ and modifying the definition of weak convergence accordingly, the definitions of limit probability measure and limit probability may be generalized [10]. The extended real line $\overline{\mathbb{R}}$ and the $\sigma$-algebra $\overline{\mathcal{R}}$ of subsets of $\overline{\mathbb{R}}$ constitute a measurable space $(\overline{\mathbb{R}},\overline{\mathcal{R}})$. The members of ${\overline{\mathscr{F}}}$ are probability measures defined on $(\overline{\mathbb{R}},\overline{\mathcal{R}})$ and induced by extended random variables (on a common sample space $\Omega$) which may take $\pm\infty$ as their values. Of course no mass will be assigned to $\{\pm\infty\}$ by any measures in ${\mathscr{F}}$. But positive probabilities can be assigned to $\{\pm\infty\}$ by measures in $\overline{\mathscr{F}}\setminus{\mathscr{F}}$. Suppose that a sequence $(\rho_{n})_{n\geq 1}$ of measures from ${\mathscr{F}}$ converges weakly to a measure $\rho\in\overline{\mathscr{F}}$. By the generalized definitions, $\rho$ is the limit probability measure of $(\rho_{n})_{n\geq 1}$ and $\rho(E)$ a limit probability where $E\in\overline{\mathcal{R}}$, even if $\rho\not\in{\mathscr{F}}$. We still write $\rho=[\lim_{n}\rho_{n}]$ to avoid confusion of limit probabilities and limits of sequences from the interval $[0,1]$. $\Box$

Example 9**.**

By the original definitions, $(\mu_{n})_{n\geq 1}$ in Example 5 has no limit probability measure because of “mass escaping to infinity”. However, according to the generalized definitions, the limit probability measure $[\lim_{n}\mu_{n}]$ of $(\mu_{n})_{n\geq 1}$ is the Dirac measure concentrated on $\{\infty\}$ induced by an extended (degenerate) random variable $Z=\infty$. As part of mass at $\{\infty\}$, $\lim_{n}[\mu_{n}(E_{n}^{\prime})]$ is not a probability. In contrast to $\lim_{n}[\mu_{n}(E_{n}^{\prime})]$, $[\lim_{n}\mu_{n}](E^{\prime})$ is a probability where $E^{\prime}={\mathbb{R}}$, see (4). In other words, calculated as the value of $[\lim_{n}\mu_{n}]$ at $E^{\prime}={\mathbb{R}}$, $[\lim_{n}\mu_{n}]({\mathbb{R}})=0$ is the limit probability corresponding to $\mu_{n}(E^{\prime}_{n})$. $\Box$

4 The Inconsistency

With a tight sequence of measures from ${\mathscr{F}}$, we can construct a sequence of measures which is also from ${\mathscr{F}}$ but not tight. The two sequences of measures are both induced by non-degenerate random variables.

Example 10**.**

In terms of random variables $X_{n}$ and $Y_{n}$ in Example 7 and Example 8, respectively, we may construct random variables $Z_{n}$ as follows.

[TABLE]

For $j\leq n$, there may be more than one $j$’s such that $X_{j}=Y_{n}$. The value of $Z_{n}$ is the largest of such $j$. Let $\mu_{n}$ be the probability measures induced by $Z_{n}$. As we can readily see, for $1\leq j<n$,

[TABLE]

and

[TABLE]

So the probability mass function of $Z_{n}$ is the same as (2) with $q$ being the probability of $\{X_{n}=0\}$ given by $\lambda_{n}$, see (5). From Example 6, $(\mu_{n})_{n\geq 1}$ is not tight, though it is constructed based on a tight sequence $(\lambda_{n})_{n\geq 1}$. $\Box$

As shown by the lemma below, an “equality” in measure-theoretic probability can be deduced, such that one side of the “equality” is a probability but the other side is not. Consider $(\lambda_{n})_{n\geq 1}$ and $(\mu_{n})_{n\geq 1}$, where $\lambda_{n}$ and $\mu_{n}$ are the probability measures induced by $X_{n}$ and $Z_{n}$, respectively (see Example 7 and (7) in Example 10).

Lemma 2**.**

The following expression

[TABLE]

is deducible, where $\lambda=[\lim_{n}\lambda_{n}]$, $\lambda(\{0\})$ is the probability of “a single Bernoulli trial fails”, and $E_{n}^{\prime}=(-\infty,n)$. But $\lim_{n}[\mu_{n}(E_{n}^{\prime})]$ is not a probability.

Proof.

For $n=1,2,\cdots$, $\{X_{n}=0\}=\{Z_{n}<n\}\cup\{Y_{n}=0\}$ and $\{Z_{n}<n\}\cap\{Y_{n}=0\}=\emptyset$, where $Y_{n}$ is given by (6). So $\{X_{n}=0\}\setminus\{Y_{n}=0\}=\{Z_{n}<n\}$ and

[TABLE]

The left-hand side of the first equality in (9) is the probability of $\{X_{n}=0\}\setminus\{Y_{n}=0\}$, i.e., “the $n^{\mathrm{th}}$ trial fails but not every trial up to the $n^{\mathrm{th}}$ one fails”. This event is characterized by $X_{n}$ and $Y_{n}$ so its probability can be calculated by $\lambda_{n}$ and $\gamma_{n}$, i.e., ${\mathbb{P}}(X_{n}=0)=\lambda_{n}(\{0\})$ and ${\mathbb{P}}(Y_{n}=0)=\gamma_{n}(\{0\})$, where $\gamma_{n}$ is the probability measure induced by $Y_{n}$. The right-hand side of the last equality in (9) is the probability of $\{Z_{n}<n\}$, i.e., “for some $j<n$ the $j^{\mathrm{th}}$ trial succeeds but the $(j+k)^{\mathrm{th}}$ trial fails for each $k=1,2,\cdots,n-j$”. This event is characterized by $Z_{n}$ and hence its probability can be calculated by $\mu_{n}$, i.e., ${\mathbb{P}}(Z_{n}<n)=\mu_{n}(E_{n}^{\prime})$, see (3). Therefore

[TABLE]

Letting $n\to\infty$ on both sides of (10) gives (8). Note that $\lim_{n}[\lambda_{n}(\{0\})]=[\lim_{n}\lambda_{n}](\{0\})=\lambda(\{0\})$ and $\lim_{n}[\gamma_{n}(\{0\})]=[\lim_{n}\gamma_{n}](\{0\})=\gamma(\{0\})=0$ where $\gamma=[\lim_{n}\gamma_{n}]$ (see Examples 7 and 8). The event associated with the probability $\lambda(\{0\})$ is $\{X=0\}$, where $X$ is a Bernoulli random variable identically distributed as $X_{n}$. However, as part of mass “escaped to infinity”, $\lim_{n}[\mu_{n}(E_{n}^{\prime})]$ is not a probability (see Examples 4 and 5). ∎

A mathematical theory is inconsistent if an expression $x=y$ is deduced in the theory, such that either (i) the expression violates the presupposition of equality, or (ii) $x$ and $y$ are numerically different. The deduced expression is erroneous only when the theory is proved to be contradiction-free as requested by Hilbert. According to Gödel’s second incompleteness theorem, measure-theoretic probability is one of the theories for which such a proof does not exist. By Lemma 2, (8) is deducible in measure-theoretic probability, which implies that the theory is not consistent as (8) violates the presupposition of equality.

Theorem 1**.**

The mathematical theory of probability (i.e., measure-theoretic probability) is inconsistent.

Proof.

Either of the two cases below shows that measure-theoretic probability is inconsistent.

(I) For probability measures induced by real-valued random variables, Lemma 1 is equivalent to its contrapositive, i.e., if $\lim_{n}[\rho_{n}^{\prime}(E_{n}^{\prime})]$ is not a probability, then at least one of the conditions (a), (b) and (c) in the lemma cannot be satisfied. Let $\lambda_{n}(\{0\})$ and $\mu_{n}(E_{n}^{\prime})$ in (10) play the roles of $\rho_{n}(E_{n})$ and $\rho_{n}^{\prime}(E_{n}^{\prime})$ in Lemma 1, respectively. By (10) and Example 8, as $n\to\infty$,

[TABLE]

So condition (a) is satisfied. As shown in Example 7, $[\lim_{n}\lambda_{n}](\{0\})$ coincides with $\lim_{n}[\lambda_{n}(\{0\})]$. Thus condition (b) is satisfied, too. By Lemma 2, (8) is deducible but $\lim_{n}[\mu_{n}(E_{n}^{\prime})]$ is not a probability. From the contrapositive of Lemma 1, condition (c) does not hold.

(II) For probability measures induced by extended random variables, we calculate the limit probabilities on both sides of (10). The limit probabilities on the two sides of (10) are $[\lim_{n}\lambda_{n}](\{0\})=q$ and $[\lim_{n}\mu_{n}](R)=0$ (see Remark 2 and Example 9), respectively. So the resultant expression is $q=0$. ∎

To defend measure-theoretic probability, someone might say: “$\lambda(\{0\})$ and $\lim_{n}[\mu_{n}(E_{n}^{\prime})]$ are the same pure number, and (8) is just a tautology derived in a theory of pure numbers.” Measure-theoretic probability is not a theory of pure numbers, however. Evidently, $\lambda(\{0\})$ is a probability, and should not be treated as a pure number. In Section 1, we have already explained why probabilities and pure numbers are quantities of different kinds.

In the literature, it has been postulated that measure-theoretic probability is a consistent theory. However, the postulated consistency of measure-theoretic probability appears to be incompatible with the presupposition of equality. Actually, it is the condition of countable additivity imposed on probability measures that leads to the contradiction. By definition, the probability measure ${\mathbb{P}}$ is $\sigma$-additive, i.e., it satisfies the assumption of countable additivity. Clearly, if the restriction of countable additivity is removed, then the inconsistency disappears. The inconsistency revealed for the $\sigma$-additive measure indicates that countable additivity is relevant to the inconsistency. Moreover, because of the assumption of countable additivity, the meaning of “a realizable event” is ambiguous. We shall clarify the ambiguity caused by countable additivity in the next section.

5 Infinite Additivity and Ideal Events

For a probability measure ${\mathbb{P}}$ and an infinite sequence $(B_{n})_{n\geq 1}$ of mutually exclusive events with ${\mathbb{P}}(B_{n})>0$ for all $n$, countable additivity requires

[TABLE]

and amounts to infinite additivity. In a nontrivial way, the assumption of infinite additivity implies that the system of events, which includes infinitely many, mutually exclusive events of positive probabilities, is closed under the formation of infinite unions and intersections. An event expressed by infinite unions or intersections is an ideal event. Relying on infinitely many operations in an infeasible experiment, ideal events will not occur in the real world. Nevertheless, such events may still be logically conceivable. Since measure-theoretic probability is sufficiently far developed, it seems that the success of the theory might be able to justify the assumption of infinite additivity [7].

No doubt measure-theoretic probability is very successful in various applications. However, the success of the theory cannot justify the assumption of infinite additivity. For an infinite sequence $(B_{n})_{n\geq 1}$ of mutually exclusive events with ${\mathbb{P}}(B_{n})>0$ for all $n$, the meaning of “$B_{n}$ is realizable” is ambiguous. As long as the probability of an event is positive, no matter how small it may be, the frequency of occurrence of the event cannot remain zero forever, and hence the event will be observed at least once in an actual experiment. In this sense the event is considered realizable. But not all of the events $B_{n},n=1,2,\cdots$ are realizable in the above sense, since the number of events corresponding to observable phenomena in any actual experiment is always finite. Therefore, even if ${\mathbb{P}}(B_{n})>0$ for all $n$ by assumption, there always exists an infinite subsequence of $(B_{n})_{n\geq 1}$, such that the occurrence of any event in the subsequence is only logically conceivable but not practically observable; otherwise there would be an instance of an infinite set observed in the real world, which is absurd. So the meaning of “$B_{n}$ is realizable” (as implied by ${\mathbb{P}}(B_{n})>0$) is not the same for all events in $(B_{n})_{n\geq 1}$. In fact, only for a finite number of events, “realizable” means “practically observable in an actual experiment”. For infinitely many events, the meaning of “realizable” is “logically conceivable but not practically observable in any actual experiment”. The ambiguity is due to the assumption of infinite additivity.

To avoid confusion due to the ambiguity, by “a realizable (or real) event” we mean “an event which is practically observable in an actual experiment”. In contrast to real events, we call an event “conceivable (or ideal)” if a positive probability is assigned to it but the event cannot be observed in any actual experiment. If one wants to acquire knowledge about the physical world by experiment, it is necessary to distinguish between real events and conceivable events; otherwise abstract properties of mathematical entities might be mistaken for properties of the physical world. As illustrated by the following two examples, abstract properties obtained by theoretical thinking are not necessarily knowledge conveyed by real events.

Example 11**.**

Consider a sequence of binomial random variables $V_{n},n=1,2,\cdots$ with probabilities

[TABLE]

According to Poisson’s theorem, as $n\to\infty$, if $p_{n}\to 0$ in such a way that $np_{n}\to c>0$, then

[TABLE]

The right-hand side of the above equality is the distribution of a Poisson random variable $V$. By definition, $V$ takes every nonnegative integer as its value with a positive probability. The Poisson distribution can be used to calculate approximately the probabilities of $\{V_{n}=k\},k=0,1,\cdots,n$ for the binomial random variables. Although a positive probability is assigned to each of $\{V=k\},k=0,1,\cdots$ for the Poisson random variable, it is impossible for all of such events $\{V=k\}$ to be observed in the physical world. $\Box$

The foundation of conventional mathematics is Zermelo-Fraenkel set theory with the axiom of choice (ZFC). As a notion in conventional mathematics, the set of nonnegative integers may be obtained by the axiom of infinity in ZFC. However, mathematical entities such as sets consisting of infinitely many elements do not exist physically and cannot be observed by experiment. No instance of such entities can be found in the real world.

Example 12**.**

Let $U$ be a continuous random variable with range ${\mathbb{R}}$ and probability density function $g$ such that $g(x)>0$ for any $x\in{\mathbb{R}}$. Random variables with normal distributions are well-known instances of such $U$. By the properties of pure numbers, on the real line, there are infinitely many disjoint intervals $I_{n}$ of positive lengths, $n=1,2,\cdots$, i.e., $\cup_{n\geq 1}I_{n}\subset{\mathbb{R}}$ and $I_{n}\cap I_{n^{\prime}}=\emptyset$ if $n\not=n^{\prime}$. From the property of the continuous random variable, for each $n$,

[TABLE]

However, only a finite number of $\{U\in I_{n}\}$ are realizable. Infinitely many of $\{U\in I_{n}\}$ are unrealizable, even though a positive probability is assigned to $\{U\in I_{n}\}$ for every $n$. $\Box$

From the point of view of pure mathematics, it is not necessary for a continuous random variable to represent any quantity in the physical world [9]. Actually, dividing a continuous physical quantity into infinitely many parts is not feasible, though it might be theoretically conceivable. However, when modeling a continuous physical quantity by a continuous random variable, some people might naively believe that, as a result of probability assignment based on the assumption of infinite additivity, infinitely many events, such as those modeled by $\{U\in I_{n}\},n=1,2,\cdots$ in Example 12, are not only conceivable but also realizable. Because of such misunderstanding, someone might even fail to differentiate abstract properties of pure numbers (which constitute a mathematical continuum) from properties of continuous physical quantities. We shall further discuss this issue and some other related issues in the next section.

6 Continuous Parameters and Microscopic Systems

The mathematical continuum is a notion in pure mathematics, and cannot be observed in realizable events. For a continuous physical quantity modeled by a continuous random variable, it is impossible to pick a precise value out of its range by measurement. On the real line, strictly between any two different pure numbers there is another. But all pure numbers are distinguishable by definition. In general, any two different points of a topological space, such as two pure numbers on the real line, can be separated in mathematical reasoning, so long as the topological space satisfies the separation axioms, which are generalizations of properties of metric spaces (e.g., see [13]). However, as shown by Example 12 in the previous section, the mathematical properties of pure numbers cannot be attached to continuous physical quantities, even though they are modeled by continuous random variables. Consequently, there is no way to measure continuous physical quantities arbitrarily accurately, regardless of whether or not the measurements are subject to the constraints imposed by the uncertainty relations in quantum mechanics.

For the same reason, if a continuous physical quantity is modeled by a continuous parameter to be adjusted for an actual experiment or measurement, and if values of the parameter are specified with an apparatus, then the apparatus cannot yield any precise value of the parameter. In fact, what can be specified with the apparatus are uncertain intervals if the parameter is real-valued, or uncertain regions if the parameter takes its values from a metric space which may not necessarily be the real line. Such uncertain intervals (or regions) are an intrinsic characteristic of modeling continuous physical quantities. As a result, it is impossible to repeat an actual experiment or measurement under the same condition described by a precisely specified value of a continuous parameter.

Example 13**.**

Let $a_{0}+\varphi$ represent a value of a continuous parameter, where $a_{0}$ is a nonnegative integer and $\varphi$ in the interval $[0,1)$. Suppose we want to specify $\varphi$ precisely, which amounts to defining a singleton set $\{\varphi\}$. Every pure number $r\in[0,1)$ has a precise expression in the form of decimal expansion

[TABLE]

Theoretically, $\{\varphi\}$ can be defined by specifying all of the digits $a_{1},a_{2},\cdots$ of $\varphi$.

[TABLE]

However, it is impossible to select any precise value from the interval $[0,1)$ in practical applications. Involving infinitely many digits, the defining property (i.e., $r_{n}=a_{n},\;n=1,2,\cdots$) of $\{\varphi\}$ cannot be used to specify $\varphi$ for an actual experiment or measurement. Only the first $m$ digits of $\varphi$, i.e., $r_{n}=a_{n},n=1,2,\cdots,m$ can be stated explicitly. The condition $r_{n}=a_{n}$ where $1\leq n\leq m$ is weaker than that required by the precise expression of $\varphi$, and defines an uncertain interval

[TABLE]

which contains $\varphi$, where $\beta_{m}=0.a_{1}a_{2}\cdots a_{m}$ and $\psi_{m}=0.a_{1}a_{2}\cdots(a_{m}+1)$. All digits $r_{m+k},k=1,2,\cdots$ of every $x\in(\beta_{m},\;\psi_{m})$ are unspecified. Since $\psi_{m}>\beta_{m}$ for any positive integer $m$, the uncertain interval is always a set with the cardinality of the continuum, though its length $\psi_{m}-\beta_{m}=10^{-m}$ can be very small for large $m$. If an actual experiment or measurement relies on specifying values of such a continuous parameter, only uncertain intervals in the range of the parameter can be specified. $\Box$

Example 14**.**

In connection with the preceding example, consider a partial sum $\sum_{1\leq n\leq m}a_{n}10^{-n}$. One may use this partial sum as an approximation to $\varphi$ in numerical calculation. Such an approximation also implies selection of a precise value $\beta_{m}$ from the interval $[0,1)$. The digits in the decimal expansion of $\beta_{m}$ are $r_{n}=a_{n}$ for $1\leq n\leq m$ and $r_{n}=0$ for $n>m$. Just like $\varphi$, the number $\beta_{m}$ cannot be specified in practical applications. As shown in Example 13, pure numbers in the interval $[0,1)$ given by their decimal expansions (or any other forms of precise expressions) are merely theoretically conceivable; they are useless for an actual experiment or measurement. No apparatus in the real world has an infinite resolution to manipulate infinitely many digits. Setting $r=0$ amounts to setting all of its digits $r_{n}=0,n=1,2,\cdots$. Such a task is not practically achievable. Similarly, for the number $\beta_{m}$ corresponding to the partial sum $\sum_{1\leq n\leq m}a_{n}10^{-n}$, it is impossible to put all the digits $r_{n}=0$ where $n=m+1,m+2,\cdots$. In other words, one cannot pick $\beta_{m}$ out of the interval $[0,1)$ with an apparatus. Since $\beta_{n}$ is an endpoint of the uncertain interval $[\beta_{m},\psi_{m})$, this example actually shows that, in practical applications, it is even impossible to specify endpoints of intervals with certainty. Generally speaking, characterized by uncertain regions, such uncertainty in modeling continuous physical quantities differs from randomness studied by probability theory. In measure-theoretic probability, by assumption, the boundary of a region in a metric space (such as the endpoints of an interval on the real line) can be specified precisely; a value of a continuous parameter may be said to lie in a region which is completely determined by its boundary; a probability may be assigned to such a region to describe randomness due to incapability of choosing a value from a continuous range. However, uncertainty in modeling continuous physical quantities cannot be captured in this way, since the boundary itself cannot be specified precisely. $\Box$

Remark 3**.**

In experiments with macroscopic systems, or in thought experiments which are not performed in the physical world, distinction between precise values and uncertain regions may be unnecessary. However, confusing uncertain regions with precisely specified values of continuous parameters may lead to undesirable consequences, such as misinterpreting random phenomena observed in actual experiments with microscopic systems. $\Box$

\begin{picture}(350.0,200.0)\par\put(250.0,180.0){\line(5,-1){60.0}} \put(250.0,180.0){\line(0,-1){60.0}} \put(310.0,168.0){\line(0,-1){60.0}} \put(250.0,120.0){\line(5,-1){60.0}} \put(280.0,113.0){\vector(0,1){80.0}} \put(280.0,97.0){\makebox(0.0,0.0)[]{Polarizer A}} \par\put(146.0,162.0){\line(4,-3){60.0}} \put(146.0,161.0){\line(-3,-5){30.0}} \put(206.0,116.0){\line(-3,-5){30.0}} \put(116.0,111.0){\line(4,-3){60.0}} \put(190.0,90.0){\vector(-4,3){80.0}} \put(159.0,112.0){\line(0,1){60.0}} \put(151.0,133.0){\makebox(0.0,0.0)[]{$\theta$}} \put(175.0,56.0){\makebox(0.0,0.0)[]{Polarizer B}} \par\put(330.0,160.0){\line(-4,-1){20.0}} \put(280.0,146.0){\line(-4,-1){85.0}} \put(159.0,112.0){\vector(-4,-1){90.0}} \par\put(220.0,25.0){\makebox(0.0,0.0)[]{Figure 1. Experiment with polarized photons.}} \end{picture}

Polarizer A

$\theta$Polarizer B

Figure 1. Experiment with polarized photons.

Example 15**.**

Photons in a linearly polarized beam produced by polarizer A arrive at polarizer B. The angle (a continuous physical quantity) between the transmission axes of the two polarizers is modeled by a continuous parameter $\theta$ (Figure 1). The intensity of the beam originating from A can be reduced so that only one photon at a time arrives at B. For a specified value of $\theta$, when arriving at B, a photon may or may not get through the polarizer. The transmission probability is $\cos^{2}\theta$ for each photon in the beam. Since the initial conditions are considered identical for all photons leaving A and striking B, according to quantum mechanics, the behavior of individual photons observed in the experiment is ontologically stochastic (e.g., see [1]).

The above quantum-mechanical interpretation implies an assumption: values of a continuous parameter (i.e., $\theta$) can be specified precisely in the experiment. But a seemingly precisely specified value of $\theta$ is actually an uncertain interval. Moreover, the direction of motion of a photon is also described by a continuous (vector-valued) parameter, and hence cannot be specified precisely. A seemingly precisely specified direction of motion for photons in the beam is actually a tiny 3-dimensional uncertain region in the range of the vector parameter. As a result, photons in the same beam may not necessarily move in the same direction. Abstract properties of mathematical entities (such as vectors) are highly idealized descriptions of the behavior of their physical counterparts (such as beams of light). For photons arriving at B, the initial conditions described by the continuous parameters cannot be specified precisely, and hence are not necessarily identical. Lack of knowledge about the initial conditions appears relevant to the observed random behavior of the photons. It is still controversial and even problematic to interpret the random behavior of the photons as a consequence of natural law. $\Box$

7 Concluding Remarks

Bernoulli trials have widespread applications to the world around us. To describe Bernoulli trials in measure-theoretic probability, one uses a sequence of probability spaces $(\Omega_{n},{\mathcal{A}}_{n},{\mathbb{P}}_{n}),n=1,2,\cdots$. For a fixed $n,(\Omega_{n},{\mathcal{A}}_{n},{\mathbb{P}}_{n})$ is called a Bernoulli scheme, which is the probabilistic model of an experiment consisting of $n$ independent trials, and each trial has two possible outcomes. Although the number of operations required by the experiment is finite, it is difficult to use such models when $n$ is large. Based on the assumption of infinite additivity, one can obtain an ideal probabilistic model $(\Omega,{\mathcal{A}},{\mathbb{P}})$ which corresponds to the limit case of the Bernoulli schemes as $n\to\infty$. The probability space $(\Omega,{\mathcal{A}},{\mathbb{P}})$ describes an ideal experiment which has infinitely many outcomes. This ideal experiment consists of an infinite number of Bernoulli trials and hence requires infinitely many operations. As we have shown, $(\Omega,{\mathcal{A}},{\mathbb{P}})$ implies the inconsistency of measure-theoretic probability, and the assumption of infinite additivity may result in confusion of ideal events and real events.

Probabilistic models of ideal experiments are very useful for calculating probabilities of real events approximately. Observed frequencies of occurrence of real events can be in good agreement with results obtained by such approximation. The use of ideal models in the above sense will obviously be unobjectionable from an empirical point of view [8]. However, conceivable events with assigned positive probabilities should not be mixed up with real events. For measure-theoretic probability, the ultimate criterion of success is whether predicted probabilities of real events in various practical applications can be verified empirically. The criterion implies a constraint: Since ideal events do not exist physically, or since it is impossible to verify the existence of ideal events in the physical world, physical existence of abstract properties derived by manipulating probabilities of ideal events should not be inferred from experiments. This constraint functions as a “firewall” of measure-theoretic probability in applications. When the constraint is satisfied, it is safe to apply measure-theoretic probability to solve problems in the real world, which may explain why measure-theoretic probability can be so successful even if it is not a consistent theory. In practice, however, violating the constraint may lead to undesirable consequences, such as mistaking a physically non-existing property of a mathematical notion for an ontological reality. A typical violation is to use experimental data to “prove” or “verify” physical existence of abstract properties of mathematical entities which have no instances in the physical world.

The inconsistency found in measure-theoretic probability appears merely in the theory, and will not affect practical applications as long as ideal events in the theory are not mistaken for observable events in the real world. Nevertheless, the inconsistency must be resolved. Physical quantities are usually modeled by mathematical entities. If one wants to measure or specify a continuous physical quantity as accurately as possible, one needs to model it by a continuous quantity in mathematics. A continuous mathematical quantity can be divided into “infinitely” many parts. But the word “infinite” has different meanings in conventional mathematics and in constructive mathematics (e.g., see [5, 12]). Measure-theoretic probability is a branch of the former. The notion of “infinite” adopted in conventional mathematics not only causes confusion of ideal events and real events, but also leads to the inconsistency eventually. Because of the assumption of infinite additivity in measure-theoretic probability, some people might mix up abstract properties of mathematical entities in theoretical thinking with properties of physical quantities observed in the real world; as a result, they might be misguided to give problematic or even wrong interpretations of random phenomena observed in experiments with microscopic systems. However, abstract properties of mathematical entities (such as infinite sets) are not necessarily properties of their physical counterparts, and hence should not be attached to physical quantities. Hopefully, the inconsistency may be resolved by treating “numbers” and “infinite” from the viewpoint of constructive mathematics.