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Let $(\Omega ,\mathcal F,\mathbb P)$ a probability space and $X:\Omega \to \mathbb R$ a random variable. Let $A$ being $\sigma (X)-$measurable. I'm having a problem to find a function $h:\mathbb R\to \mathbb R$ s.t. $$h(X(\omega ))=1_A(\omega ).$$

In somehow, I would like to set $$h(x)=1_{X(A)}(x),$$ but the problem it's that it can happen that $X^{-1}(X(A))\supsetneq A$, and thus, we can have $X(\omega )\in X(A)$ but $\omega \notin A$. So, is there a way to define $h$ properly or not really ?

Walace
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  • There exists a set $B\subset\mathbb R$ measurable such that $X^{-1}(B)=A$. So set $h(x)=1_B(x)$. – Shashi May 02 '20 at 19:58
  • This result is called factorization lemma (or Doob-Dynkin lemma), see e.g. here https://en.m.wikipedia.org/wiki/Doob%E2%80%93Dynkin_lemma – saz May 02 '20 at 20:09
  • @saz: Thanks, I didn't know this lemma. Sorry to ask you that, but I saw that you are an expert in brownian motion, stochastic calculus... I had a lot of unanswerered question in this area. Would you kindly check them (in case it can be interesting) ? For example this one ? And others, see my activities in my profile if you have time. Thanks a lot for everything. :) – Walace May 02 '20 at 20:22

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