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.

posted almost 3 years ago

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