Suppose that all functions ${f_n},f$ are integrable. Is $\lim_{n \rightarrow \infty} \int f_n(x)dx = \int f(x)dx?$

Question

Let ${f_n}$ be a sequence of continuous, strictly positive functions on $\mathbb{R}$ which converges uniformly to the function $f.$ Suppose that all the functions ${f_n},f$ are integrable. Is $$\lim_{n \rightarrow \infty} \int f_n(x)dx = \int f(x)dx\ ?$$

This is clearly true if on an interval of $\mathbb{R}$ because then there is $k$ and $\epsilon$ such that $f_k$ and $\epsilon$ which dominates the other $f_n$. Any suggestions? Thanks (Its a qual problem from a previous exam)

Answer 1

nope. let's get rid of the assumption 'strictly' positive - we'll just add our nonnegative functions to a strictly positive and integrable functions of your choice $f$. the point is to make $\int f_n dx$ divergent. what we do is we let $$f_n := f + \frac{1}{n}\chi_{[0, n^2]}$$ where $\chi$ is the characteristic function. obviously the integrals get bigger and diverge to infinity while we still have uniform convergence, since $||f - f_n||_{\infty} = 1/n$

edit: sorry, to preserve continuity just add something nicer than the characteristic function - it's easy to 'smooth' it out, but the general idea stays the same