interchanging integral and limit

Question

Consider the limit $$\lim_{n\to \infty }\int_0^1(1-t^n)f(t)\;dt$$ Where $f$ is a continuous function on $[0,1]$.

Is it safe to put the limit inside the integral and so that case the limit would be $\int_0^1 f(t)\;dt$? are there any conditions to verify here before putting the limit inside ?

https://math.stackexchange.com/q/253696/344419 – Andrew Li Apr 17 '18 at 20:32 — Andrew Li, Apr 17 '18 at 20:32
It is safe when you have a proof of it. – zhw. Apr 17 '18 at 21:04 — zhw., Apr 17 '18 at 21:04

score 0 · Answer 1 · answered Apr 17 '18 at 20:33

0

You can use the Dominated convergence Theorem, your function is dominated by $f(t)$.

answered Apr 17 '18 at 20:33

Dalamar

767

score 0 · Answer 2 · answered Apr 17 '18 at 20:34

Since $f$ is continuous on $[0,1]$, there is a finite maximum $M=\max_{t\in[0,1]}|f(t)|$. Then $|(1-t^n)f(t)|\leq M$ for all $t\in[0,1]$ and all $n$ (i.e. the sequence of functions $(1-t^n)f(t)$ is dominated by the constant function $M$), so Lebesgue's Dominated Convergence Theorem gives you

$$\lim_{n\to\infty}\int_0^1(1-t^n)f(t)\,dt=\int_0^1\lim_{n\to\infty}(1-t^n)f(t)\,dt=\int_0^1f(t)\,dt.$$

interchanging integral and limit

2 Answers2