# Dominated convergence exercise

Given a function $$f \in L^1(\mathbb{R})$$, suppose I want to evaluate the following integral

$$\lim_{n\to\infty} \int_{\mathbb{R}}e^{-nx^2}f(x)dx$$

It looks to me like one could use dominated convergence here. If we let $$f_n(x) = e^{-nx^2}f(x)$$, then clearly $$|f_n(x)| \leq |f(x)|$$ for $$n \geq 0$$, and we know that $$|f(x)|$$ is an integrable function. The issue I am having is showing that $$f_n(x)$$ itself is integrable - it seems clear to me that it would be, but not sure how to show it (maybe it just follows from the above inequality since $$|f(x)|$$ is integrable?)

Also, suppose we can show dominated convergence theorem applies, the limit is a little bit interesting as it depends on $$x$$. Clearly, if $$x\neq 0$$, then we have

$$\lim_{n\to\infty} e^{-nx^2}f(x) = 0,$$ but if $$x = 0$$, then what happens?

Thanks!

• $\{0\}$ is of measure $0$, so it doesn't matter what happens there. – saulspatz Dec 4 '18 at 19:13

$$f_n$$ is integrable because $$|f_n| \le |f|$$, as you mentioned.
We have $$\lim_{n \to \infty} f_n(x) = \begin{cases} 0 & x \ne 0 \\ f(0) & x = 0. \end{cases}$$
The integral of this function is simply zero. So if you show that you can use the dominated convergence theorem, then $$\lim_{n \to \infty} \int f_n = 0$$.
• What else needs to be shown to use dominated convergence? I thought it was that the sequence $f_n$ had to be bounded above by an integrable function and be integrable themselves and then conditions were satisfied. – Sorey Dec 4 '18 at 19:17