I found a problem in the ISI JRF sample question paper.The question is as follows:
If $f_1,f_2,...,f\geq 0$ are Lebesgue integrable on $\mathbb R$ such that $\lim\limits_{n\to \infty}\int_{-\infty}^y f_n(x)dx=\int_{-\infty}^y f(x)dx$ for every $y\in \mathbb R\cup \{+\infty\}$.Show that $\liminf\limits_{n\to \infty}\ \int_U f_n(x)dx\geq \int_U f(x)dx$ for any open subset $U\subset \mathbb R$.
Here is my attempt:
Take any open interval $(a,b)$ with $-\infty\leq a<b\leq +\infty$,then we can show that $\lim\limits_{n\to \infty}\int_a^b f_n(x)dx=\int_a^b f(x)dx$.This follows due to the fact that $\int_a^b=\int_{-\infty}^b-\int_{-\infty}^a$.Now take any open subset $U\subset \mathbb R$ then we have a result that $U$ can be written as a disjoint union of countably many open intervals.So we write $U=\bigcup\limits_{n=1}^\infty(a_n,b_n)$ which is a countable disjoint union of open intervals.Let $U_m=\bigcup\limits_{k=1}^m (a_k,b_k)$.Now, $\int_{U_m} f_n(x)dx=\sum\limits_{k=1}^m \int_{a_k}^{b_k} f_n(x)dx$ and $\int_{U_m} f(x)dx=\sum\limits_{k=1}^m\int_{a_k}^{b_k} f(x)dx$.Now note that $U_m\uparrow U$ as $m\to \infty$.Since $f_n$ and $f$ are measurable non-negative functions,so $\nu_n(E)=\int_E f_n(x)dx$ and $\nu(E)=\int_E f(x)dx$ are measures on $\mathbb R$.Now measures are continuous with respect to increasing limits which gives $\nu_n(U_m)\uparrow \nu_n(U)$ as $m\to \infty$ which can be rewritten as $\lim\limits_{m\to \infty} \int_{U_m} f_n(x)dx=\int_U f_n(x)dx$.In a similar way, $\lim\limits_{m\to \infty} \int_{U_m} f(x)dx=\int_U f(x)dx$.But for a fixed $m$,we have $\lim\limits_{n\to \infty}\sum\limits_{k=1}^m \int_{a_k}^{b_k} f_n(x)dx=\sum\limits_{k=1}^m\int_{a_k}^{b_k} f(x)dx$ which implies that $\lim\limits_{n\to \infty}\int_{U_m} f_n(x)dx=\int_{U_m} f(x)dx$.Now we apply limit as $m\to \infty$ on both sides,to get $\lim\limits_{m\to \infty}\lim\limits_{n\to \infty}\int_{U_m} f_n(x)dx=\lim\limits_{m\to \infty}\int_{U_m} f(x)dx=\int_U f(x)dx$.Now on the left hand side we can interchange the limits without any problem as the integral inside the interated limit is non-negative.So we have $\lim\limits_{n\to \infty}\lim\limits_{m\to \infty}\int_{U_m} f_n(x)dx=\int_U f(x)dx$ and this implies that $\lim\limits_{n\to \infty} \int_U f_n(x)dx=\int_U f(x)dx$.
I am not sure whether my solution is correct because even if it is correct I have deduced something stronger than demanded in the question.That's why I have a doubt whether I did it correctly or there is some error hidden in this solution.Can anyone check if it is okay?