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3 votes

Null function and Lebesgue measure on $\mathbb{R}^N$

This is an application of Tonelli's theorem. Namely, suppose $(X, \mu)$ and $(Y, \nu)$ are $\sigma$-finite measure spaces and $A\subset X\times Y$ is a measurable subset with the property that for $\...
Jonathan Hole's user avatar
2 votes

Image of measurable sets under one to one (a.e) functions

That would depend heavily on the measure. In an extreme case, if your measure $\mu$ is concentrated on a singleton then the condition "one-to-one a.e." is satisfied trivially and the ...
Lieven's user avatar
  • 971
2 votes

A conjecture about measurable functions

Your conjecture is false. Let $\mathcal H$ be the $\sigma$-algebra of Lebesgue measurable sets in $\mathbb R$. Knowing whether each half-line $(-\infty, x)$ occurs, i.e. whether $X < x$, uniquely ...
Robert Israel's user avatar
2 votes

When is measurability of a function $f$ equivalent to $f$ being almost everywhere the limit of measurable functions?

Yes, the theorem is correct. Let us prove it: $\def\AAA {\mathcal{A}} \def\BBB {\mathcal{B}} \def\NN {\mathbb{N}}$ Theorem: let $(X,\AAA,\mu)$ be a complete measure space, and $(Y,\BBB)$ a ...
Ramiro's user avatar
  • 17.8k

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