Let $f_n:\mathbb{R}\rightarrow [0, 1]$ be functions such that $\sup_{x \in \mathbb{R}}f_n(x) = 1/n$ and $||f||_1 = 1$.

Question

Let $f_n:\mathbb{R}\rightarrow [0, 1]$ be functions such that $\sup_{x \in \mathbb{R}}f_n(x) = 1/n$ and $||f_n||_1 = 1$. Set $F(x) = \sup_{n \in \mathbb{N}}f_n(x)$.

Prove that $\int_\mathbb{R}F(x)dx = \infty.$

I am not really sure what tools to use in this question. My guess is its just some trick I am not seeing and isn't all that complex. Any suggestions? Thanks.

Answer 1

Suppose that $\int_{\mathbb{R}}F(x)dx<\infty$ and note that $f_{n}\le F$ and that $f_{n}$ converge pointwise to $0$. This means we could apply dominated convergence which would mean:

$1=\lim_{n\to\infty}\lvert\lvert f_{n}\rvert\rvert_{\mathbb{R}}=0$

which is a contradiction.