1. ## Measure theory questions

Hi, I have a couple of things I am stuck on.

1.
Use the dominated convergence theorem to find
$\displaystyle \lim_{n\to\infty}\int_{1}^{\infty}f_{n}(x) dx$

where
$\displaystyle f_{n}(x)=\frac{\sqrt{x}}{1+nx^3}$

I deduced that since the sequence $\displaystyle \{f_{n}\}$ converges to 0 pointwise and we have dominating function $\displaystyle x^{-2.5}$ such that $\displaystyle \frac{\sqrt{x}}{1+nx^3} \leq x^{-2.5}$.
All I need to do now to show that the dominating function is integrable so that the sequence of the integrand converges to 0.
But how do you show this?

2. How do you show that a sequence $\displaystyle \{x_{n}\}$ that converges to $\displaystyle x$ in $\displaystyle \mathbb{R}$ then $\displaystyle f\chi_{(-\infty,x_{n}]} \to f\chi_{(-\infty,x]}$

Cheers

2. For 1)
I suggest you try dominating the sequence with$\displaystyle f(x)=\frac{1}{x^2}$

To check integrability you just need to compute

$\displaystyle \int_{[1,\infty]} f d\lambda = \int_{1}^\infty \frac{1}{x^2}dx$

since f is continuous the lebesgue integral and riemann integral are the same and you can compute it.