Lebesgue Integral help

Question

I'm studying for my Real Analysis Masters Exam and the only thing that I don't understand at all is Lebesgue integrals and cant find any good examples. A old master exam problem I'm trying says:

Prove $\lim_{n \to \infty} \int_{\mathbb{R}} \frac{\cos(nx)}{1+nx^2} \,d\lambda (x) = 0$

where this is the Lebesgue integral.

My thought is to create a sequence of functions $f_n = \frac{\cos(nx)}{1+nx^2} $ and then use the Dominate Convergence Theorem to bring the limit inside, though I'm not sure how to show I meet the criteria or if I'm even on the right track.

Any tips or help would be greatly appreciated!

Answer 1

The idea is indeed to use the dominated convergence theorem.

For any $x\in\mathbb{R}$, we have that $$|f_n(x)|=\left|\dfrac{\cos(nx)}{1+nx^2}\right|\leq \dfrac{1}{1+nx^2}\to 0,$$

which tells us that $f_n$ converge pointwise to $0$.

Continuing with the bound, $$|f_n(x)|=\left|\dfrac{\cos(nx)}{1+nx^2}\right|\leq \dfrac{1}{1+nx^2}\leq \dfrac{1}{1+x^2}$$

But $g(x)=\dfrac{1}{1+x^2}$ verifies that $$\displaystyle\int_\mathbb{R}|g(x)|dx=\displaystyle\int_\mathbb{R}g(x)dx=\pi<\infty,$$

so $g$ is integrable and the Dominated convergence theorem assures that $\displaystyle\int_\mathbb{R} f_n(x) dx\to 0.$

Answer 2

Notice that $$\left | \frac{cos(nx)}{1+nx^{2}} \right |\leq \left | \frac{1}{1+nx^{2}} \right |\leq \left | \frac{1}{1+x^{2}} \right |, \ \ \forall n\in \mathbb{N}, \forall x\in \mathbb{R}.$$ You have that the right hand side is a measurable function which integral over $\mathbb{R}$ is finite. Then, you have the conditions to apply the Dominate Convergence Theorem and conclude what you want.