I have a sequence of functions: for $p\geq 0$ and on $\mathbb R$, consider $$f_n(x)=n^p1_{(0,\frac{1}{n})}$$ I want to show:
- $f_n\to f$ pointwise
- For which $p$ is $\lim_{n\to\infty}\int f_ndm=0?$ ($m$ denoting the Lebesgue Measure.)
- Find a dominating function $g$ such that $f_n(x)\leq g(x)$ for all $n$, and use the Dominated Convergence Theorem to interpret the answer in (b) in light of $g$.
My solution:
1)Let $x\in \mathbb R, \epsilon >0$. Then choose $N$ such that $x>\frac{1}{N}$. Then for any $n>N$, $x>\frac{1}{N}>\frac{1}{n}$, and $\lvert f_n(x)\rvert=0<\epsilon$. Thus $f_n\to f$ pointwise.
2) $$\lim_{n\to\infty}\int f_ndm=\lim_{n\to\infty}\int n^p1_{(0,\frac{1}{n})}(x)=\lim_{n\to\infty}n^pm((0,\frac{1}{n}))=\lim_{n\to\infty}n^p\frac{1}{n}=\lim_{n\to\infty}n^{p-1}$$ Now, since we want this limit to go to $0$, $p<1$ must be true since $n\in\mathbb N$.
3) For #3, I'm not really sure how to find a function that dominates this sequence of functions as the ones I come up with depend on $n$, but I believe $g$ must not depend on $n$ for the use of DCT. I would like some help on coming up with an example.
Are my solutions correct? And I would like some help on coming up with a dominating function for the given sequence.
$x^{-p}$ is a dominating integrable function for $p<1$. Note that $0<x<\frac 1 n$ implies $n^{p} <x^{-p}$.