let $\{f,f_{n}:n\ge 1\}$ be $\mathbb R$-valued measurable function on $(\Omega,\mathcal A,\mu)$

(a) Assume that $f_{n}\uparrow f$ and that there exist an $(\mathcal A,\mathcal B_{\mathbb R})$ measurable function such that $\int h d\mu< \infty $ and $f_{n}\ge h\quad \forall n$. Then Show that $\int f_{n} d\mu\uparrow \int f d\mu$.

(b) show by counter example that the above hypothesis cannot be dropped.

thanks for help


(a) Hint: use monotone convergence theorem for the non-negative sequence $(f_n-h)_{n\geqslant 1}$.

(b) Take $f_n(x):=-\frac 1n\chi_{(-n,0)}$: this increases to $0$, but the integral is $1$ for each $n$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.