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I found this in another language. I am wondering if anyone could find this in a English book please. Or if anyone know how to prove this please. Preferably one could tell me a reference book. Thank you very much. Let $(\Omega, F, P)$ be a probability space. and let $g$ and $h $ be functions such that $\int_A g,dP\leqslant \int_A h,dP $ for all $A \in F$, then for $g,h \in \mathbb{L}^1(P)$, $g\leqslant h$

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  • Interesting question, but it's off topic in this forum; try math.stackexchange.com instead. By the way, you can help others help you by showing your own work on the problem. – Robert Dodier May 16 '21 at 23:35

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You will need to use measure theory. If it is the opposite case, you can find a contradiction.

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