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I'm trying to grasp the intuition of conditional expectations. On a probability space $(\Omega,\mathcal{F},\mathbb{P})$, let $\mathcal{G}\subset\mathcal{F}$ be a sub-sigma-algebra. Denote $X:\Omega\to\mathbb{R}$ to be a random variable. If we have a Lipschitz continuous function $f:\mathbb{R}\to\mathbb{R}$ with Lipschitz constant $C$, is it true that $$ \left|\mathbb{E}[f(X):\mathcal{G}] - f(0)\right| \leq C \mathbb{E}\left[||X||:\mathcal{G}\right] $$ holds?

So in the above, both sides a random variables. So I just applied the definition of Lipschitz continuity and I am unsure if the above is valid since we are conditioning on a sub-sigma-algebra.

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  • $\begingroup$ Conditional expectation inherit the monotonicity from the normal expectation. $\endgroup$ Oct 4, 2017 at 10:48
  • $\begingroup$ Thank you quallenjager $\endgroup$
    – jerom
    Oct 4, 2017 at 10:58

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Since $\left\lvert \mathbb E\left[Y\mid\mathcal G \right] \right\rvert\leqslant \mathbb E\left[\left\lvert Y\right\rvert\mid\mathcal G \right] $ for any integrable random variable $Y$, we have $$ \left\lvert \mathbb E\left[f(X)\mid\mathcal G \right] -f(0) \right\rvert =\left\lvert \mathbb E\left[f(X)-f(0) \mid\mathcal G \right] \right\rvert \leqslant \mathbb E\left[\left\lvert f(X)-f(0)\right\rvert \mid\mathcal G \right].$$ If $Y_1$ and $Y_2$ are two integrable random variables such that $Y_1\leqslant Y_2$ almost surely, then $\mathbb E\left[Y_1\mid\mathcal G \right] \leqslant \mathbb E\left[Y_2\mid\mathcal G \right]$ almost surely. Apply this to $Y_1:=\left\lvert f(X)-f(0)\right\rvert$ and $Y_2:=C\left\lvert X\right\rvert $.

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