User:Boris Tsirelson/sandbox1

Criticism

A quote from [Dur, Sect. 1.4(c), p. 33]:

$(S,\mathcal S)$ is said to be nice if there is a 1-1 map $\phi$ from $S$ into $\R$ so that $\phi$ and $\phi^{-1}$ are both measurable.

Such spaces are often called standard Borel spaces, but we already have too many things named after Borel. The next result shows that most spaces arising in applications are nice.

(4.12) Theorem. If $S$ is a Borel subset of a complete separable metric space $M$, and $\mathcal S$ is the collection of Borel subsets of $S$, then $(S,\mathcal S)$ is nice.

References

[I]	Kiyosi Itô, "Introduction to probability theory", Cambridge (1984).   MR0777504   Zbl 0545.60001