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In a proof of Jensen's Inequality that I am reading, the following is used:

  • If for a real valued random variable $X$, we have $X(\omega)<\beta$, then $\mathbb{E}[X]\leq \beta$.
  • Why can we deduce from the strict inequality of the arguments only a non strict inequality for the expectation ?. ( later on it is being shown by some more arguments that the inequality involving the expectation in fact turns out to be strict but some work has to be done to get there ).
  • So what property of the expectation or integral is it that in the first instance only allows us to deduce a non-strict inequality ?.
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  • $\begingroup$ Hello. Can you show us the exact statement of Jensen's inequality, as well as the proof, and ideally tell us the source? It's hard to say otherwise. In general it often happens that limiting processes preserve non-strict inequalities but not strict inequalities. In this case, there is a standard argument that shows $X < \beta$ implies $\mathbb E[X] < \beta$: see here, but I think this is "less obvious" than the fact that expectations preserve non-strict inequalities. Is that what the "more work" looks like? $\endgroup$
    – Izaak van Dongen
    Commented Jul 8 at 19:20

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