# Decomposition random variable with conditional expectation [closed]

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.

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

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