Let $f$ a measurable function and $X$ a random variable. Is the following equality always true?
$$\mathbb E[f(X)|f(X)] = \mathbb E[f(X)|X] $$
2026-04-01 17:03:33.1775063013
Conditional means and measurable functions
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Yes. $f(X)$ is measurable with respect to $\sigma(f(X))$ and hence also with respect to $\sigma(X)$ since $\sigma(f(X))\subseteq \sigma(X)$. In particular, both conditional expectations are just $f(X)$.