Convergence of Lebesgue integrals under monotonicity conditions.

Question

Suppose one has two sequences of positive measurable functions such that $\int f_{n}(x)dx \le \int g_{n}(x)dx$ for all $n$.

Prove that if $\lim f_{n}(x)=f(x)$ and

$\lim g_{n}(x)=g(x)$ then $\int f(x)dx \le \int g(x)dx$.

Answer 1

Counterexample.

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Let $h$ be a positive measurable function with $\int h(x)dx=1$

Let $g_n(x)=\mathbf1_{[n,n+1]}(x)+h(x)$ so that $g(x)=h(x)$ and $\int g(x)dx=\int h(x)dx=1$.

Let $f_n(x)=g_1(x)$ for every $n$ so that $f(x)=g_1(x)+h(x)$ and $\int f(x)dx=2$.

Then $f_n,g_n$ are positive measurable functions with $\int f_n(x)dx=2=\int g_n(x)dx$ for all $n$.

However: $$\int f(x)dx=2>1=\int g(x)dx$$