Limit of $f_n(x)=f(x)(1+(sin x)^n)$

Question

I am asked to find

$\lim_{n\rightarrow \infty}\int_{\mathbb{R}}f_nd\lambda$, where $f_n(x)=f(x)(1+(sin(x))^n)$

using a convergence theorem (Monotone Convergenve Theorem, Dominated Convergence Theorem or Fatou's lemma), given that $\lambda$ is the Lebesgue measure and $f:\mathbb{R}\rightarrow\mathbb{R}$ is integrable.

So what I've done so far is defining an integrable function $g(x)=3f(x)$ for which I know $|f_n(x)|\leq g(x)$ $ \forall n\in\mathbb{N}$. However, I got stuck at finding the pointwise limit of $f_n(x)$. In particular, consider $x=k*0.5\pi$ and all the other points in $\mathbb{R}$. These limits are not the same as $n\rightarrow \infty$, so I can't use the DCT anymore.

My biggest concern is the limit of $f_n$. Could somebody please explain.

Thanks in advance!

Answer 1

Your $g$ is a good start.

Now note that $\lim_{n\to\infty}f_n=f$ a.e. (almost everywhere: possibly not equal on $\{\frac{\pi}{2}+n\pi\ |\ n\in\mathbb{Z}\}$, which is a null set), so by the a.e.-version DCT we obtain $$\lim_{n\rightarrow \infty}\int_{\mathbb{R}}f_nd\lambda=\int_{\mathbb{R}}fd\lambda$$