Evaluate Integrals Using Convergence Theorems

Question

I'm trying to evaluate these integrals using Convergence Theorems, but I'm not really sure how to go about it. Here are the integrals:

• For $\phi$ bounded and continuous, $\psi\in L^1(m)$, $\lim_{n\to\infty}\int_{\mathbb R}\phi(x/n)\psi(x)dm(x)$
• For $\phi$ continuous and compactly supported, $\psi\in L^1(m)$, $\lim_{n\to\infty}\int_{-\infty}^\infty \phi(nx)\psi(x)dm(x)$
• $\lim_{n\to\infty}\int_{-\infty}^\infty \frac{n}{x}\sin(x/n)e^{-\lvert x\rvert}dm(x)$
• $\lim_{n\to\infty}\int_{[0,1]}(1+nx^2)(1+x^2)^{-n}dm(x)$

$m$ denotes the Lebesgue measure, and $L^1(m)$ denotes the set of integrable functions with respect to the Lebesgue measure.

I think for the first two, I need to find a dominating function for the integrand and then find the limit of the integrand, and for the last two, I believe I need to show that they are monotone increasing and find the limit of the integrand, but I haven't been able to come up with dominating functions/proof that the sequence of functions are monotone increasing.

I would really love to get some help.

Answer 1

Here are some hints:

1. Use the fact that $\phi$ is bounded, say $|\phi|\le M$ and $M\psi$ as a dominating function. Apply dominated convergence.

2. Similar idea to 1. Since $\phi$ is compactly supported, if $x \ne 0$, what is the value of $\phi(nx)$ for large $n$?

3. Apply 1. to $\phi(x) = \sin x/x$ and use the fact that $\sin x/x\to 1$ as $x\to 0$.

4. Notice that the denominator $(1+x^2)^n \ge 1 + nx^2 + n(n-1)x^4/2$ by the binomial theorem holds for each $n$. Find an appropriate dominating function.