Hello,

I am looking at a measure space $\displaystyle (X,\mathcal{A},\mu)$ where $\displaystyle u \in \mathcal{L}^{1}(\mu)$. Put $\displaystyle K_{n}= \left\{ \vert u \vert \leq n \right\}$ for every $\displaystyle n \geq 1$ and let $\displaystyle u_{n}=u\mathbb{1}_{K_{n}}$. I am asked to show that:

$\displaystyle \lim \int_{K_{n}} u d\mu = \int_{X} u d\mu$

$\displaystyle \lim \int_{X\setminus K_{n}} \vert u \vert d\mu = 0$

The set $\displaystyle K_{n}$ consists of real-values $\displaystyle \mu$-integrable functions whose positive parts are less than or equal to $\displaystyle n$, i.e $\displaystyle K_{n}= \left\{x \in X \mid \vert u(x) \vert \leq n \right\}$

This means that $\displaystyle K_{n} \subseteq X \subseteq \mathcal{A}$.

No sure how to start. Hints would be appreciated.