I'm stuck on the following problem from Bass's "Real Analysis for Graduate Students" (Exercise 7.21):
Suppose $\mu_n$ is a sequence of measures on $(X,\mathcal{A})$ such that $\mu_n(X)=1$ for all $n$ and $\mu_n(A)$ converges as $n\to\infty$ for each $A\in\mathcal{A}$. Call the limit $\mu(A)$.
(1) Prove that $\mu$ is a measure.
(2) Prove that $\int f\;d\mu_n\to\int f\;d\mu$ whenever $f$ is bounded and measurable.
(3) Prove that $$\int f\;d\mu\le\liminf_{n\to\infty}\int f\;d\mu_n$$ whenever $f$ is non-negative and measurable.
Note that this question is from the chapter that introduces the monotone convergence theorem, Fatou's lemma, and the dominated convergence theorem. I was able to prove (1) fairly easily. However, I'm stuck on (2). I'm trying to show that $$\left|\int f\;d\mu_n-\int f\;d\mu\right|\to0$$ but can't seem to find an appropriate bound. Perhaps it would be better to show that
$$\int f\;d\mu\le\liminf\int f\;d\mu_n$$
and
$$\limsup\int f\;d\mu_n\le\int f\;d\mu$$
but I doubt the author intended for us to use (3) to prove (2), otherwise he'd switch the order. Any help with (2) or (3) is much appreciated.
Best Answer
Note that (3) is for $f$ nonnegative not-necessarily bounded above, so you will need something to go from (2) to (3).
For (2), note that it is obviously true for simple functions $f$. So pick an arbitrary bounded measurable $f$, and squeeze it between two simple functions $u_k$ and $l_k$: $$ f-2^{-k}\leq l_k\leq f\leq u_k\leq f+2^{-k} $$ Integrate with respect to $\mu_n$ $$ \int_X l_k\,\mathrm{d}\mu_n\leq\int_X f\,\mathrm{d}\mu_n\leq \int_X u_k\,\mathrm{d}\mu_n $$ so $$ \int_X l_k\,\mathrm{d}\mu\leq\lim_{n\to\infty}\int_X f\,\mathrm{d}\mu_n\leq \int_X u_k\,\mathrm{d}\mu $$ But we also have, by monotonicity of integral, $$ \int_X l_k\,\mathrm{d}\mu\leq\int_X f\,\mathrm{d}\mu\leq \int_X u_k\,\mathrm{d}\mu $$ So $$ \left\lvert\int_X f\,\mathrm{d}\mu-\lim_{n\to\infty}\int_X f\,\mathrm{d}\mu_n\right\rvert\leq\int_X (u_k-l_k)\,\mathrm{d}\mu\leq 2^{1-k} $$ But $k$ is arbitrary.
Can you see how to use (2) to prove (3)?