Is it impossible to prove Jensen's inequality by way of step functions?
Jensen's Inequality: Let $\varphi:\Bbb R\to \Bbb R$ be convex, and $f:[0,1]\to\Bbb R$ be integrable, and suppose $\varphi\circ f$ is integrable over [0,1]. Then $$ \varphi\left(\int_{[0,1]} f\right)\le \int_{[0,1]}\varphi\circ f $$ A proof from step functions:
I have seen a proof of this inequality for every step function f.
My question is, can we then extend this proof to all measurable f satisfying the assumptions of the proof?
I've encountered a number of other proofs of this theorem, but I'm particularly interested to see how one might do this from the result for step functions. If we knew that $\varphi$ were continuous then I would consider trying to use one of the several convergence theorems for the Lebesgue integral. But since we don't assume continuity then this seems like a bad route.
But if we are going to extend from the result for step functions, it seems like we must use $f = \lim_{n\to\infty} s_n$ for a sequence of step functions. But if the limit can't pass through the function $\varphi$ then it seems to me that there simply cannot be any way to infer the full result from the result for step functions.
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The following users marked this post as Works for me:
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whybecause | (no comment) | Feb 11, 2022 at 21:48 |
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