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 $\varphi$ from $S$ into $\R$ so that $\varphi$ and $\varphi^{-1}$ 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.

It is not specified in the definition, whether $\varphi(S)$ must be a Borel set, or not. The proof of the theorem provides just a Borel 1-1 map $\varphi:S\to\R$ without addressing measurability of the function $\varphi^{-1}$ and the set $\varphi(S)$. (A complete proof would be considerably harder.) Later, in the proof of Theorem (1.6) of [Dur, Sect. 4.1(c)], measurability of $\varphi^{-1}$ and $\varphi(S)$ is used (see the last line of the proof).

References

[Dur]

Richard Durrett, "Probability: theory and examples", second edition, Duxbury Press (1996).   MR1609153