Operators on the Stopping Time Space

TL;DR

This paper characterizes operators on the stopping time space that preserve a copy of it using associated measures, showing preservation is equivalent to the non-separability of a measure set.

Contribution

It introduces a measure-based criterion for identifying operators that preserve a copy of the stopping time space, linking operator properties to measure-theoretic non-separability.

Findings

Operators preserve a copy of $S^1$ iff associated measures are non-separable.

Measures $\mu_{x^{**}} ext{ characterize operator preservation properties.

Provides a measure-theoretic criterion for operator analysis on $S^1$.

Abstract

Let $S^1$ be the stopping time space and $\mathcal{B}_1(S^1)$ be the Baire-1 elements of the second dual of $S^1$. To each element $x^{**}$ in the space $\mathcal{B}_1(S^1)$ we associate a positive Borel measure $\mu_{x^{**}}$ on the Cantor set. We use the measures $\{\mu_{x^{**}}: x^{**}\in\mathcal{B}_1(S^1) \}$ to characterize the operators $T:X\to S^1$, defined on a space $X$ with an unconditional basis, which preserve a copy of $S^1$. In particular, we show that $T$ preserves a copy of $S^1$ if and only if the set $\{\mu_{x^{**}}:\;x^{**}\in\mathcal{B}_1(S^1)\}$ is non separable as a subset of $\mathcal{M}(2^\mathbb{N})$.