probability , expectations in $L^1$

Question

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 $g,h \in \mathbb{L}^1(P)$, $g\leqslant h$

Answer 1

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