Fubini Type Theorems for the strong McShane and strong Henstock-Kurzweil integrals

TL;DR

This paper establishes Fubini type theorems for strong McShane and Henstock-Kurzweil integrals of Banach space-valued functions on two-dimensional intervals, extending integral theory in Banach spaces.

Contribution

It proves Fubini theorems for these integrals, which was previously unexplored for Banach space-valued functions in this context.

Findings

Fubini theorems for strong McShane integrals

Fubini theorems for strong Henstock-Kurzweil integrals

Extension of integral theory to Banach space-valued functions

Abstract

In this paper, we will prove Fubini type theorems for the strong McShane and strong Henstock-Kurzweil integrals of Banach spaces valued functions defined on a closed non-degenerate interval $[a,b] =[a_{1}, b_{1}] \times [a_{2}, b_{2}] \subset \mathbb{R}^{2}$.

Equations

$$[a,b]=[a_1,b_1]\times [a_2,b_2],$$

$${\mathcal{I}}_{[a,b]}={\mathcal{I}}_{[a_1,b_1]}\times {\mathcal{I}}_{[a_2,b_2]},$$

$$F(I\cup J)=F(I)+F(J)$$

$$\sum _{(t,I)\in P}\Vert f(t)|I|-F(I)\Vert <\epsilon$$

$$F(I)=(M){\int }_{I}f\left(F(I)=(HK){\int }_{I}f\right),\text{ for all }I\in {\mathcal{I}}_{[a,b]},$$

$$\mathcal{S}\mathcal{H}\mathcal{K}(K)\subset \mathcal{H}\mathcal{K}(K),\mathcal{SM}(K)\subset \mathcal{M}(K),\mathcal{M}(K)\subset \mathcal{H}\mathcal{K}(K),\mathcal{SM}(K)\subset \mathcal{S}\mathcal{H}\mathcal{K}(K).$$

$$\mathcal{S}\mathcal{H}\mathcal{K}(K)=\mathcal{H}\mathcal{K}(K)\text{ and }\mathcal{SM}(K)=\mathcal{M}(K).$$

$$\sum _{(t,I)\in P}\sum _{(s,J)\in Q}||f(t)-f(s)||.|I\cap J|<\epsilon$$

$${\mathcal{S}}^{\ast }\mathcal{H}\mathcal{K}(K)\subset \mathcal{S}\mathcal{H}\mathcal{K}(K),{\mathcal{S}}^{\ast }\mathcal{M}(K)=\mathcal{SM}(K),\mathcal{SM}(K)=\mathcal{B}(K),$$

$$f_{t_1}(s_2)=f(t_1,s_2),\text{ for all }{s}_2\in [a_2,b_2]$$

$$f_{t_2}(s_1)=f(s_1,t_2),\text{ for all }{s}_1\in [a_1,b_1],$$

$$(M){\int }_{[a,b]}w\text{mathbbm}{1}_Z=0,$$

$$(M){\int }_{[a,b]}\text{mathbbm}{1}_Z=0,$$

$$\left|\sum _{(t,I)\in P_n}w(t)\text{mathbbm}{1}_Z(t)|I|-0\right|=\sum _{(t,I)\in P_n}w(t)\text{mathbbm}{1}_Z(t)|I|<\frac{\epsilon }{(n+1)2^{n+1}}$$

$$A_n=\left\{t\in [a,b]:\frac{1}{n+1}\le w(t)<\frac{1}{n}\right\},\text{ for all }n\in \mathbb{N},$$

$$\delta(t)=\left\{\begin{array}[]{ll}\delta_{n}(t)&\text{if }t\in A_{n},~{}n=0,1,2,\dotsc\\ 1&\text{if }t\in B_{0}\\ \end{array}\right.$$

$$\begin{array}{rl}\left|\sum _{(t,I)\in P}\text{mathbbm}{1}_Z(t)|I|-0\right|=& \sum _{\underset{t\in B_0}{(t,I)\in P}}\text{mathbbm}{1}_Z(t)|I|+\sum _{\underset{t\in A_0}{(t,I)\in P}}\text{mathbbm}{1}_Z(t)|I|+\sum _{n=1}^{+\infty }\sum _{\underset{t\in A_n}{(t,I)\in P}}\text{mathbbm}{1}_Z(t)|I|\\ =& \sum _{\underset{t\in A_0}{(t,I)\in P}}\text{mathbbm}{1}_Z(t)|I|+\sum _{n=1}^{+\infty }(n+1)\sum _{\underset{t\in A_n}{(t,I)\in P}}\frac{1}{n+1}\text{mathbbm}{1}_Z(t)|I|\\ \le & \sum _{\underset{t\in A_0}{(t,I)\in P}}w(t)\text{mathbbm}{1}_Z(t)|I|+\sum _{n=1}^{+\infty }(n+1)\sum _{\underset{t\in A_n}{(t,I)\in P}}w(t)\text{mathbbm}{1}_Z(t)|I|\\ <& \frac{\epsilon }{2}+\sum _{n=1}^{+\infty }\frac{\epsilon }{2^{n+1}}=\frac{\epsilon }{2}+\frac{\epsilon }{2}=\epsilon .\end{array}$$

$$\left|\sum _{(t,I)\in P_n}\text{mathbbm}{1}_Z(t)|I|-0\right|=\sum _{(t,I)\in P_n}\text{mathbbm}{1}_Z(t)|I|<\frac{\epsilon }{n{2}^n}$$

$$B_n=\left\{t\in [a,b]:n-1\le ||f(t)||<n\right\},\text{ for all }n\in \mathbb{N}.$$

$$\delta (t)={\delta }_n(t)$$

$$\begin{array}{rl}\sum _{(t,I)\in P}\Vert f(t)\text{mathbbm}{1}_Z(t)|I|-\theta \Vert \le & \sum _{n=1}^{+\infty }\sum _{\underset{t\in B_n}{(t,I)\in P}}||f(t)||\text{mathbbm}{1}_Z(t)|I|\\ <& \sum _{n=1}^{+\infty }n\sum _{\underset{t\in B_n}{(t,I)\in P}}\text{mathbbm}{1}_Z(t)|I|<\sum _{n=1}^{+\infty }\frac{\epsilon }{2^n}=\epsilon .\end{array}$$

$$f_{t_1}\in \mathcal{SM}[a_2,b_2]\text{ and }{f}_{t_2}\in \mathcal{SM}[a_1,b_1].$$

$$(M){\int }_{[a_1,b_1]}\left((M){\int }_{[a_2,b_2]}{f}_{t_1}\right)=(M){\int }_{[a,b]}f.$$

$$(M){\int }_{[a_2,b_2]}\left((M){\int }_{[a_1,b_1]}{f}_{t_2}\right)=(M){\int }_{[a,b]}f.$$

$$\begin{array}{r}\sum _{(t,I)\in P}\Vert f(t)|I|-(M){\int }_{I}f\Vert <\frac{\epsilon }{2}\end{array}$$

$$\begin{array}{r}\sum _{(t_2,I_2)\in Q_2^{(t_1)}}\Vert f_{t_1}(t_2)|I_2|-(M){\int }_{I_2}{f}_{t_1}\Vert <\frac{\epsilon }{2(1+(b_1-a_1))}\end{array}$$

$${\delta }_2^{(t_1)}(t_2)\le \delta (t_1,t_2),\text{ for all }{t}_2\in [a_2,b_2].$$

$${\delta }_1(t_1)=\min \left\{{\delta }_2^{(t_1)}(t_2):(t_2,I_2)\in Q_2^{(t_1)}\right\},\text{ for all }{t}_1\in [a_1,b_1]$$

$$Q=\left\{((t_1,t_2),I_1\times I_2):(t_1,I_1)\in Q_1\text{ and }(t_2,I_2)\in Q_2^{(t_1)}\right\}$$

$$\begin{array}{rl}& \sum _{(t_1,I_1)\in Q_1}\Vert g(t_1)|I_1|-(M){\int }_{I_1\times [a_2,b_2]}f\Vert \le \\ \le & \sum _{(t_1,I_1)\in Q_1}\left(|I_1|\Vert (M){\int }_{[a_2,b_2]}{f}_{t_1}-\sum _{(t_2,I_2)\in Q_2^{(t_1)}}{f}_{t_1}(t_2)|I_2|\Vert +\Vert \sum _{(t_2,I_2)\in Q_2^{(t_1)}}f(t_1,t_2)|I_1|.|I_2|-(M){\int }_{I_1\times [a_2,b_2]}f\Vert \right)\end{array}$$

Full text

1. Introduction and Preliminaries

The Fubini theorem belongs to the most powerful tools in Analysis. It establishes a connection between the so called double integrals and repeated integrals. Theorem X.2 in [19] is the Fubini theorem for Bochner integral of Banach spaces valued functions defined on the Cartesian product $U\times V$ of Euclidean spaces $U$ and $V$. Here, we will prove a Fubini type theorem for the strong McShane integral of Banach spaces valued functions defined on two-dimensional compact intervals, see Theorem 2.4. Henstock in [6] proved a Fubini–Tonelli type theorem for the Perron integral of real valued functions defined on two-dimensional compact intervals. Tuo-Yeong Lee in [17] and [18] proved several Fubini–Tonelli type theorems for the Henstock–Kurzweil integral of real valued functions defined on $m$-dimensional compact intervals in terms of the Henstock variational measures. In this paper, we will prove a Fubini type theorem for the strong Henstock–Kurzweil integral of Banach spaces valued functions defined on two-dimensional compact intervals, see Theorem 2.5.

Throughout this paper $X$ denotes a real Banach space with its norm $||\cdot||$. The Euclidean space $\mathbb{R}^{2}$ is equipped with the maximum norm $||\cdot||_{\infty}$. $B_{2}(t,r)$ denotes the open ball in $\mathbb{R}^{2}$ with center $t=(t_{1},t_{2})\in\mathbb{R}^{2}$ and radius $r>0$. $A^{o}$, $\partial A$ and $|A|$ denote, respectively, the interior, boundary and Lebesgue measure of a subset $A\subset\mathbb{R}^{2}$. For any two vectors $a=(a_{1},a_{2})$ and $b=(b_{1},b_{2})$ with $-\infty<a_{i}<b_{i}<+\infty$, for $i=1,2$, we set

[TABLE]

which is said to be a closed non-degenerate interval in $\mathbb{R}^{2}$. By $\mathcal{I}_{[a,b]}$ the family of all closed non-degenerate subintervals in $[a,b]$ is denoted. It is easy to see that

[TABLE]

where $\mathcal{I}_{[a_{1},b_{1}]}$ ($\mathcal{I}_{[a_{2},b_{2}]}$) is the family of all closed non-degenerate subintervals in $[a_{1},b_{1}]$ ($[a_{2},b_{2}]$). A pair $(t,I)$ of an interval $I\in\mathcal{I}_{[a,b]}$ and a point $t\in[a,b]$ is called an $\mathcal{M}$-tagged interval in $[a,b]$, $t$ is the tag of $I$. Requiring $t\in I$ for the tag of $I$ we get the concept of an $\mathcal{HK}$-tagged interval in $[a,b]$. A finite collection $P$ of $\mathcal{M}$-tagged intervals ($\mathcal{HK}$-tagged intervals) in $[a,b]$ is called an $\mathcal{M}$-partition ($\mathcal{HK}$-partition) in $[a,b]$, if $\{I\in\mathcal{I}_{[a,b]}:(t,I)\in P\}$ is a collection of pairwise non-overlapping intervals in $\mathcal{I}_{[a,b]}$. Two closed non-degenerate intervals $I,J\in\mathcal{I}_{[a,b]}$ are said to be non-overlapping if $I^{o}\cap J^{o}=\emptyset$. A positive function $\delta:[a,b]\to(0,+\infty)$ is said to be a gauge on $[a,b]$. We say that an $\mathcal{M}$-partition ($\mathcal{HK}$-partition) $P$ in $[a,b]$ is

• •

$\mathcal{M}$-partition ($\mathcal{HK}$-partition) of $[a,b]$, if $\bigcup_{(t,I)\in P}I=[a,b]$,

• •

$\delta$-fine if for each $(t,I)\in P$, we have $I\subset B_{2}(t,\delta(t)).$

A function $F:\mathcal{I}_{[a,b]}\to X$ is said to be an additive interval function if

[TABLE]

for any two non-overlapping intervals $I,J\in\mathcal{I}_{[a,b]}$ with $I\cup J\in\mathcal{I}_{[a,b]}$.

Definition 1.1**.**

A function $f:[a,b]=[a_{1},b_{1}]\times[a_{2},b_{2}]\to X$ is said to be strongly McShane integrable (strongly Henstock-Kurzweil integrable) on $[a,b]$, if there is an additive interval function $F:\mathcal{I}_{[a,b]}\to X$ such that for every $\varepsilon>0$ there exists a gauge $\delta$ on $[a,b]$ such that

[TABLE]

for every $\delta$-fine $\mathcal{M}$-partition ($\mathcal{HK}$-partition) $P$ of $[a,b]$.

By Theorem 3.6.5 in [20], if $f$ is strongly McShane integrable (strongly Henstock-Kurzweil integrable) on $[a,b]$, then $f$ is McShane integrable (Henstock-Kurzweil integrable) on $[a,b]$ and

[TABLE]

where $F$ is the additive interval function from the definition of strong integrability.

Let $K$ be a compact non-degenerate interval in an Euclidean space. We denote by $\mathcal{SM}(K)$ ($\mathcal{SHK}(K)$) the set of all strongly McShane (strongly Henstock-Kurzweil) integrable functions defined on $K$ with $X$-values. If $\mathcal{M}(K)$ ($\mathcal{HK}(K)$) is the set of all McShane (Henstock-Kurzweil) integrable functions defined on $K$ with $X$-values, then

[TABLE]

If the Banach space $X$ is finite dimensional, then we obtain by Proposition 3.6.6 in [20] that

[TABLE]

Definition 1.2**.**

A function $f:[a,b]\to X$ has the property $\mathcal{S^{*}M}$ ($\mathcal{S^{*}HK}$) if for every $\varepsilon>0$ there is a gauge $\delta$ on $[a,b]$ such that

[TABLE]

for each pair of $\delta$-fine $\mathcal{M}$-partitions ($\mathcal{HK}$-partitions) $P$ and $Q$ of $[a,b]$.

We denote by $\mathcal{S^{*}M}(K)$ ($\mathcal{S^{*}HK}(K)$) the set of all functions defined on $K$ with $X$-values having the property $\mathcal{S^{*}M}$ ($\mathcal{S^{*}HK}$). Clearly, $\mathcal{S^{*}M}(K)\subset\mathcal{S^{*}HK}(K)$. By Lemma 3.6.11, Theorem 3.6.13 and Theorem 5.1.4 in [20], we have

[TABLE]

where $\mathcal{B}(K)$ is the set of all Bochner integrable functions defined on $K$ with $X$-values.

The basic properties of the McShane and Henstock-Kurzweil integrals can be found in [20], [14], [15], [16], [1], [3], [21], [2], [4], [5], [7]-[9] and [11]-[13].

Given a function $f:[a,b]\to X$, for each $t_{1}\in[a_{1},b_{1}]$ and $t_{2}\in[a_{2},b_{2}]$ we define $f_{t_{1}}:[a_{2},b_{2}]\to X$ and $f_{t_{2}}:[a_{1},b_{1}]\to X$ by setting

[TABLE]

and

[TABLE]

respectively.

2. The Main Results

The main results are Theorems 2.4 and 2.5. Let us start with a few auxiliary lemmas, which will be formulated for the case of the McShane integral ($\mathcal{SM}[a,b]$, $\mathcal{S^{*}M}[a,b]$) but all of them hold for the Henstock-Kurzweil integral ($\mathcal{SHK}[a,b]$, $\mathcal{S^{*}HK}[a,b]$) as well. It suffices to check their proofs with the necessary replacement of $\mathcal{M}$-partitions by $\mathcal{HK}$-partitions, etc.

Lemma 2.1**.**

Let $Z\subset[a,b]=[a_{1},b_{1}]\times[a_{2},b_{2}]$. Then the following statements hold.

• (i)

Let $w:[a,b]\to[0,+\infty)$ be such that $w(t)>0$ for each $t\in Z$. If $w\mathbbm{1}_{Z}$ is McShane integrable on $[a,b]$ with

[TABLE]

then $\mathbbm{1}_{Z}$ is McShane integrable on $[a,b]$ with $(M)\int_{[a,b]}\mathbbm{1}_{Z}=0.$

• (ii)

Let $f:[a,b]\to X$. If $\mathbbm{1}_{Z}$ is McShane integrable on $[a,b]$ with

[TABLE]

then $f\mathbbm{1}_{Z}\in\mathcal{SM}[a,b]$ with $(M)\int_{[a,b]}f\mathbbm{1}_{Z}=\theta,$ where $\theta$ is the zero vector in $X$.

Proof.

$(i)$ Let $\varepsilon>0$ be given. By Lemma 3.4.2 in [20], for each $n=0,1,2,\dotsc$ there exists a gauge $\delta_{n}$ on $[a,b]$ such that

[TABLE]

whenever $P_{n}$ is a $\delta_{n}$-fine $\mathcal{M}$-partition in $[a,b]$. We set

[TABLE]

$A_{0}=\{t\in[a,b]:w(t)\geq 1\}$ and $B_{0}=\{t\in[a,b]:w(t)=0\}$. Since $B_{0}\cap Z=\emptyset$ it follows that $\mathbbm{1}_{Z}(t)=0$ for every $t\in B_{0}$. Define a gauge $\delta$ on $[a,b]$ as follows

[TABLE]

and let $P$ be a $\delta$-fine $\mathcal{M}$-partition of $[a,b]$. Then we obtain by (2.1) that

[TABLE]

This means that $\mathbbm{1}_{Z}$ is McShane integrable on $[a,b]$ with $(M)\int_{[a,b]}\mathbbm{1}_{Z}=0$.

$(ii)$ Let $\varepsilon>0$ be given. By Lemma 3.4.2 in [20], for each $n\in\mathbb{N}$ there exists a gauge $\delta_{n}$ on $[a,b]$ such that

[TABLE]

whenever $P_{n}$ is a $\delta_{n}$-fine $\mathcal{M}$-partition in $[a,b]$. We set

[TABLE]

Define a gauge $\delta$ on $[a,b]$ so that

[TABLE]

whenever $n\in\mathbb{N}$ and $t\in B_{n}$. If $P$ is a $\delta$-fine $\mathcal{M}$-partition of $[a,b]$, then we obtain by (2.2) that

[TABLE]

This means that $f\mathbbm{1}_{Z}$ is strongly McShane integrable on $[a,b]$ with $(M)\int_{[a,b]}f\mathbbm{1}_{Z}=\theta$ and the proof is finished. ∎

Lemma 2.2**.**

Assume that $f\in\mathcal{SM}[a,b]$ and for each $t_{1}\in[a_{1},b_{1}]$ and $t_{2}\in[a_{2},b_{2}]$, we have

[TABLE]

Then the following statements hold.

• (i)

The function $g(t_{1})=(M)\int_{[a_{2},b_{2}]}f_{t_{1}}$, for all $t_{1}\in[a_{1},b_{1}]$, is strongly McShane integrable on $[a_{1},b_{1}]$ and

[TABLE]

• (ii)

The function $h(t_{2})=(M)\int_{[a_{1},b_{1}]}f_{t_{2}}$, for all $t_{2}\in[a_{2},b_{2}]$, is strongly McShane integrable on $[a_{2},b_{2}]$ and

[TABLE]

Proof.

Let $\varepsilon>0$ be given. Then, since $f$ is strongly McShane integrable on $[a,b]$ there exists a gauge $\delta$ on $[a,b]$ such that

[TABLE]

for each $\delta$-fine $\mathcal{M}$-partition $P$ of $[a,b]$.

Since $f_{t_{1}}$ is strongly McShane integrable on $[a_{2},b_{2}]$ whenever $t_{1}\in[a_{1},b_{1}]$, there exists a gauge $\delta^{(t_{1})}_{2}$ on $[a_{2},b_{2}]$ such that

[TABLE]

for each $\delta^{(t_{1})}_{2}$-fine $\mathcal{M}$-partition $Q^{(t_{1})}_{2}$ of $[a_{2},b_{2}]$. We can choose each $\delta^{(t_{1})}_{2}$ so that

[TABLE]

We now define a gauge $\delta_{1}$ on $[a_{1},b_{1}]$ by setting

[TABLE]

and let $Q_{1}$ be a $\delta_{1}$-fine $\mathcal{M}$-partition of $[a_{1},b_{1}]$. Then

[TABLE]

is a $\delta$-fine $\mathcal{M}$-partition of $[a,b]$. We have

[TABLE]

and since

[TABLE]

we obtain by (2.3) and (2.4) that

[TABLE]

This means that $g$ is strongly McShane integrable on $[a_{1},b_{1}]$ and $(i)$ holds.

Since the proof of $(ii)$ is similar to that of $(i)$, the lemma is proved. ∎

Lemma 2.3**.**

Let $f\in\mathcal{S^{*}M}[a,b]$, let

[TABLE]

and $Z=(Z_{1}\times[a_{2},b_{2}])\cup([a_{1},b_{1}]\times Z_{2})$.

Then the following statements hold.

• (i)

$\mathbbm{1}_{Z_{1}}$* is Mcshane integrable on $[a_{1},b_{1}]$ with $(M)\int_{[a_{1},b_{1}]}\mathbbm{1}_{Z_{1}}=0$.*

• (ii)

$\mathbbm{1}_{Z_{2}}$* is Mcshane integrable on $[a_{2},b_{2}]$ with $(M)\int_{[a_{2},b_{2}]}\mathbbm{1}_{Z_{2}}=0$.*

• (iii)

$\mathbbm{1}_{Z}$* is Mcshane integrable on $[a,b]$ with $(M)\int_{[a,b]}\mathbbm{1}_{Z}=0$.*

Proof.

$(i)$ By virtue of Theorem 3.6.13 in [20], for each $t_{1}\in Z_{1}$ there exists $w(t_{1})>0$ with the following property: for each gauge $\delta^{(t_{1})}_{2}$ on $[a_{2},b_{2}]$ there exist a pair of $\delta^{(t_{1})}_{2}$-fine $\mathcal{M}$-partitions $Q^{(t_{1})}_{1},Q^{(t_{1})}_{2}$ of $[a_{2},b_{2}]$ such that

[TABLE]

If we choose $w(t_{1})=0$ at all $t_{1}\in[a_{1},b_{1}]\setminus Z_{1}$, then $w=w\mathbbm{1}_{Z_{1}}$. Thus, if $w$ is McShane integrable on $[a_{1},b_{1}]$ with

[TABLE]

then we obtain by $(i)$ in Lemma 2.1 that $(i)$ holds.

Since $f\in\mathcal{S^{*}M}[a,b]$, given $\varepsilon>0$ there exists a gauge $\Delta$ on $[a,b]$ such that

[TABLE]

for each pair of $\Delta$-fine $\mathcal{M}$-partitions $Q_{1},Q_{2}$ of $[a,b]$.

Note that for each $t_{1}\in[a_{1},b_{1}]$ the function $\Delta^{(t_{1})}_{2}:[a_{2},b_{2}]\to(0,+\infty)$ defined by

[TABLE]

is a gauge on $[a_{2},b_{2}]$. Then, by (2.5) for each $t_{1}\in Z_{1}$, we can choose a pair of $\Delta^{(t_{1})}_{2}$-fine $\mathcal{M}$-partitions $P^{(t_{1})}_{1},P^{(t_{1})}_{2}$ of $[a_{2},b_{2}]$ such that

[TABLE]

For each $t_{1}\in[a_{1},b_{1}]\setminus Z_{1}$, we choose a $\Delta^{(t_{1})}_{2}$-fine $\mathcal{M}$-partition $P^{(t_{1})}$ of $[a_{2},b_{2}]$ and set $P^{(t_{1})}_{1}=P^{(t_{1})}_{2}=P^{(t_{1})}$. In this case, it easy to see that (2.7) holds also.

We now define a gauge $\Delta_{1}$ on $[a_{1},b_{1}]$ by setting

[TABLE]

and let $\pi$ be a $\Delta_{1}$-fine $\mathcal{M}$-partition of $[a_{1},b_{1}]$. Since

[TABLE]

and

[TABLE]

are $\Delta$-fine $\mathcal{M}$-partitions of $[a,b]$, we obtain by (2.7) and (2.6) that

[TABLE]

This means that $w$ is McShane integrable on $[a_{1},b_{1}]$ with $(M)\int_{[a_{1},b_{1}]}w=0$.

The proof of $(ii)$ is similar to that of $(i)$.

$(iii)$ Since $\mathbbm{1}_{Z_{1}}$ is Mcshane integrable on $[a_{1},b_{1}]$ with $(M)\int_{[a_{1},b_{1}]}\mathbbm{1}_{Z_{1}}=0$, given $\varepsilon>0$ there exists a gauge $\delta_{1}$ on $[a_{1},b_{1}]$ such that

[TABLE]

for each $\delta_{1}$-fine $\mathcal{M}$-partition $\pi$ of $[a_{1},b_{1}]$.

We now define a gauge $\delta$ on $[a,b]$ by setting

[TABLE]

and let $P$ be a $\delta$-fine $\mathcal{M}$-partition of $[a,b]$. There exists a finite collection $\mathcal{D}_{P}$ of pairwise non-overlapping intervals in $[a_{2},b_{2}]$ such that

• •

$[a_{2},b_{2}]=\bigcup_{I\in\mathcal{D}_{P}}I$,

• •

for each $I\in\mathcal{D}_{P}$ there exists $P^{(I)}\subset P$ such that

[TABLE]

and

[TABLE]

Hence, for each $I\in\mathcal{D}_{P}$ the collection

[TABLE]

is a $\delta_{1}$-fine $\mathcal{M}$-partition of $[a_{1},b_{1}]$. Note that

[TABLE]

and

[TABLE]

Therefore, we obtain by (2.8) that

[TABLE]

This means that $(iii)$ holds and the proof is finished. ∎

We are now ready to present the first main result.

Theorem 2.4**.**

Let $f\in\mathcal{SM}[a,b]$, let

[TABLE]

$Z=(Z_{1}\times[a_{2},b_{2}])\cup([a_{1},b_{1}]\times Z_{2})$* and $f_{0}=f.\mathbbm{1}_{[a,b]\setminus Z}$.*

Then the following statements hold.

• (i)

$f_{0}\in\mathcal{SM}[a,b]$* and for each $(t_{1},t_{2})\in[a_{1},b_{1}]\times[a_{2},b_{2}]$, we have $(f_{0})_{t_{1}}\in\mathcal{SM}[a_{2},b_{2}]$ and $(f_{0})_{t_{2}}\in\mathcal{SM}[a_{1},b_{1}]$.*

• (ii)

The function

[TABLE]

is strongly McShane integrable on $[a_{1},b_{1}]$ and

[TABLE]

• (iii)

The function

[TABLE]

is strongly McShane integrable on $[a_{2},b_{2}]$ and

[TABLE]

Proof.

$(i)$ Since $\mathcal{SM}[a,b]=\mathcal{S^{*}M}[a,b]$, we obtain by Lemma 2.3 that the function $\mathbbm{1}_{Z}$ is McShane integrable with $(M)\int_{[a,b]}\mathbbm{1}_{Z}=0$. Hence, by $(ii)$ in Lemma 2.1 we have $f\mathbbm{1}_{Z}\in\mathcal{SM}[a,b]$ and $(M)\int_{[a,b]}f\mathbbm{1}_{Z}=\theta$. It follows that

[TABLE]

We now fix an arbitrary $t_{1}\in[a_{1},b_{1}]$. There are two cases to consider.

• (a)

$t_{1}\in Z_{1}$. In this case, we have $(f_{0})_{t_{1}}(t_{2})=\theta$ at all $t_{2}\in[a_{2},b_{2}]$. Thus, $(f_{0})_{t_{1}}\in\mathcal{SM}[a_{2},b_{2}]$.

• (b)

$t_{1}\not\in Z_{1}$. In this case, we have $f_{t_{1}}\in\mathcal{SM}[a,b]$ and $(f_{0})_{t_{1}}(t_{2})=f_{t_{1}}(t_{2})-f_{t_{1}}(t_{2})\mathbbm{1}_{Z_{2}}(t_{2})$ at all $t_{2}\in[a_{2},b_{2}]$. Lemma 2.3 together with Lemma 2.1 yields that $f_{t_{1}}.\mathbbm{1}_{Z_{2}}\in\mathcal{SM}[a_{2},b_{2}]$ with $(M)\int_{[a_{2},b_{2}]}f_{t_{1}}.\mathbbm{1}_{Z_{2}}=\theta$. Therefore, $(f_{0})_{t_{1}}\in\mathcal{SM}[a_{2},b_{2}]$.

Hence, we have $(f_{0})_{t_{1}}\in\mathcal{SM}[a_{2},b_{2}]$ for each $t_{1}\in[a_{1},b_{1}]$. Similarly, it can be proved that $(f_{0})_{t_{2}}\in\mathcal{SM}[a_{1},b_{1}]$ for each $t_{2}\in[a_{2},b_{2}]$.

Therefore, Lemma 2.2 together with $(i)$ and (2.9) yields that $(ii)$ and $(iii)$ hold, and this ends the proof. ∎

Theorem 2.5**.**

Let $f\in\mathcal{S^{*}HK}[a,b]$, let

[TABLE]

$Z=(Z_{1}\times[a_{2},b_{2}])\cup([a_{1},b_{1}]\times Z_{2})$* and $f_{0}=f\mathbbm{1}_{[a,b]\setminus Z}$.*

Then the following statements hold.

• (i)

$f_{0}\in\mathcal{SHK}[a,b]$* and for each $(t_{1},t_{2})\in[a_{1},b_{1}]\times[a_{2},b_{2}]$, we have $(f_{0})_{t_{1}}\in\mathcal{SHK}[a_{2},b_{2}]$ and $(f_{0})_{t_{2}}\in\mathcal{SHK}[a_{1},b_{1}]$.*

• (ii)

The function

[TABLE]

is strongly Henstock-Kurweil integrable on $[a_{1},b_{1}]$ and

[TABLE]

• (iii)

The function

[TABLE]

is strongly Henstock-Kurweil integrable on $[a_{2},b_{2}]$ and

[TABLE]

Proof.

$(i)$ By Lemma 2.3 the function $\mathbbm{1}_{Z}$ is Henstock-Kurweil integrable with $(HK)\int_{[a,b]}\mathbbm{1}_{Z}=0$. Hence, by $(ii)$ in Lemma 2.1 we have $f\mathbbm{1}_{Z}\in\mathcal{SHK}[a,b]$ and $(HK)\int_{[a,b]}f\mathbbm{1}_{Z}=\theta$. It follows that

[TABLE]

We now fix an arbitrary $t_{1}\in[a_{1},b_{1}]$. There are two cases to consider.

• (a)

$t_{1}\in Z_{1}$. In this case, we have $(f_{0})_{t_{1}}(t_{2})=\theta$ at all $t_{2}\in[a_{2},b_{2}]$. Thus, $(f_{0})_{t_{1}}\in\mathcal{SHK}[a_{2},b_{2}]$.

• (b)

$t_{1}\not\in Z_{1}$. In this case, we have $f_{t_{1}}\in\mathcal{SHK}[a,b]$ and $(f_{0})_{t_{1}}(t_{2})=f_{t_{1}}(t_{2})-f_{t_{1}}(t_{2})\mathbbm{1}_{Z_{2}}(t_{2})$ at all $t_{2}\in[a_{2},b_{2}]$. Lemma 2.3 together with Lemma 2.1 yields that $f_{t_{1}}.\mathbbm{1}_{Z_{2}}\in\mathcal{SHK}[a_{2},b_{2}]$ with $(HK)\int_{[a_{2},b_{2}]}f_{t_{1}}.\mathbbm{1}_{Z_{2}}=\theta$. Therefore, $(f_{0})_{t_{1}}\in\mathcal{SHK}[a_{2},b_{2}]$.

Hence, we have $(f_{0})_{t_{1}}\in\mathcal{SHK}[a_{2},b_{2}]$ for each $t_{1}\in[a_{1},b_{1}]$. Similarly, it can be proved that $(f_{0})_{t_{2}}\in\mathcal{SHK}[a_{1},b_{1}]$ for each $t_{2}\in[a_{2},b_{2}]$.

Therefore, Lemma 2.2 together with $(i)$ and (2.10) yields that $(ii)$ and $(iii)$ hold, and this ends the proof. ∎