Uniform convergence and integrability of a sequence

Question

I have these two theorems but I will mainly focus on the second:

$1.$ Suppose that $\{F_n\}$ converges uniformly to $F$ on $S=[a,b]$. Assume that $F$ and all $F_n$ are integrable on $[a,b]$. Then $$\star\int_{a}^{b}F(x) \ dx= \lim_{n \to \infty}\int_{a}^{b}F_n(x) \ dx.$$

and

$2.$ Suppose that $\{F_n\}$ converges pointwise to $F$ and each $F_n$ is integrable on $[a,b]$.

$a)$ If the convergence is uniform, then $F$ is integrable on $[a,b]$ and $\star$ holds.

$b)$ If the sequence $\{\parallel F_n \parallel _{[a,b]}\}$ is bounded and $F$ is integrable on $[a,b]$, then $\star$ holds.

Now can anyone give me examples where the second theorem works, my book does not give me exmaples. BTW I am new to this stuff.

Answer 1

Consider $F_n(x)=(1-x)^n$ for $x\in[0,1]$, a bounded function sequence that converges pointwise to: \begin{align} F(x)= \begin{cases} 1, \hspace{15pt} x = 0\\ 0, \hspace{15pt} x\in(0,1] \end{cases} \end{align} The integral: \begin{align} \int^1_0F(x)\mathrm d x =0\end{align} And: \begin{align} \int^1_0 F_n(x)\mathrm d x=\int^1_0(1-x)^n\mathrm d x= \frac{1}{n+1}\end{align}