Prove "Measure theoretic" Fatous lemma. Let $f\geq0$ Prove that $\mu(x | \liminf f_n(x)>t)\leq\liminf \mu(x | f_n(x)>t)$ for each $t>0$
I think this is an easy consequence of fatous lemma. We have the following equalitiew $\mu(x | \liminf f_n(x)>t)=\int_X\chi(x)_{\liminf f_n(x)>t}$ and $\liminf \mu(x | f_n(x)>t)=\liminf\int_X\chi(x)_{f_n(x)>t}$. Thus employing fatous lemma we see that all we have to show is that$\chi(x)_{\liminf f_n(x)>t}\leq \liminf \chi(x)_{f_n(x)>t}$. This is not hard to show. all we have to do is show that if LHS is 1, then so is the RHS. If $\liminf f_n(x)>t$ then for large enough $n$ we have $f_n(x)>t$ for all $n$ and so $\liminf \chi(x)_{f_n(x)>t}=1$ as desired,
Is this correct?