If for any arbitrary function $\phi(\cdot)$ $$E(\phi(X))=E(\phi(Y)),$$ then $X=Y$ a.s.
Can somebody name this theorem?
2026-03-31 18:26:50.1774981610
Expected value theorem
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If $E[\phi(X)]=E[\phi(Y)]$ for every $\phi$, the most you can conclude is that $X$ and $Y$ have the same distribution. This follows from taking $\phi(x) = I(x\in B)$ as $B$ ranges over an appropriately large collection of sets.
To see why $X=Y$ need not be true, even if $X$ and $Y$ are defined on the same space, consider $X$ uniformly distributed on $[0,1]$ and take $Y:=1-X$.