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Why given some information set $I$, any random variable $x_t$ can be decomposed into the sum:

$$x_t = E(x_t | I) + v_t $$ where $E(v_t | I) = 0$.

I'm looking for a clear proof.

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closed as unclear what you're asking by user158565, Michael Chernick, Jeremy Miles, kjetil b halvorsen, Siong Thye Goh May 16 at 14:59

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  • $\begingroup$ Would not work for Cauchy distribution. $\endgroup$ – Carl May 16 at 0:40