Sequence of functions and integral multiple choice question

Question

Define $f_n(x)= 1 $, if $x\in [-n-2,-n)$ and $f_n(x)=0$ otherwise. Let $\alpha=\int_{-\infty}^\infty \lim_{n\rightarrow \infty} f_n(x) \, dx$ and $\beta=\lim_{n\rightarrow \infty} \int_{-\infty}^\infty f_n(x) \, dx$. then

1. $0<\alpha <1, ~\beta=1$
2. $\alpha=0,~\beta=\infty$
3. $\alpha=\beta=0$
4. $\alpha=0,~\beta=2$

We may be required to check the uniform convergence of this sequence. I am not able to do much

Answer 1

Fix $x$. Then for large enough $n$, $x \notin [-n-2,n) $ so $f_n(x) = 0$. Consequently, for all $x$, $\lim_{n \to \infty} f_n(x) = 0$. Thus, $\alpha = \int_{-\infty}^\infty 0dx = 0$.

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Fix $n$, then $\int_{-\infty}^\infty f_n(x) \,dx = \int_{-n-2}^{-n} 1\,dx = -n - (-n-2) = 2$. Thus, $\beta = \lim_{n \to \infty} 2 = 2$.

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The above serves as a counterexample to the exchange of limit and integral. The reason you cannot apply DCT is because any function pointwise dominating all the $f_n$ is positive and bounded away from $0$ hence not integrable on $\mathbb R$.