Find the limit of the integral

Question

I need help to solve this question, I think I should use the dominated convergence theorem.

1. In the space ([0,1],B([0,1]),$\lambda$) (where $\lambda$ is the measure of lebesgue restricted to the interval [0,1]), we defined the succesion of functions f${_n}$:[0,1]$\rightarrow$$\mathbb{R}$

f${_n}$: min$\left( \frac{e^{-nx^{2}}}{\sqrt{x}}, n \right)$

Find:$\lim_{n \rightarrow {\infty}}\int_0^{1}f_n \: d\lambda$

Thanks in advance.

Answer 1

Show:

1. $f_n \to 0$ a.e.

2. With $g(x)=\frac{1}{\sqrt{x}}$ we have: for each $n$: $|f_n| \le g$ a.e.

Now use the dominated convergence theorem.