A characterization of an almost everywhere constant variable

**ymar** · Mar 7th 2011, 09:04 AM

I've been thinking for several hours now, and can't see it.

Let $([0,1],\mathcal{M},\mathbb{P})$ be a probability space. Let $f:[0,1]\rightarrow\mathbb{R}$ be a $\mathbb{P}$ -integrable variable (in the Lebesgue sense). Then the following conditions are equivalent:

- $f$ is $\mathbb{P}$ -almost everywhere constant
- $e^{\int_{[0,1]}f(x)\mathrm{d}\mathbb{P}}=\int_{[0,1]}e^{f(x)}\mathrm{d}\mathbb{P}$

Obviously I must be missing some inequality, probably well known. Any help will be appreciated.

**theodds** · Mar 7th 2011, 03:39 PM

This is an attainment of Jensen's Inequality.

**ymar** · Mar 7th 2011, 11:12 PM

Thank you very much.

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