An application of the dominated convergence theorem to approximate a function.

Question

In "Measure theory and probability theory (pag. 58)" by Krishna and Soumendra we can found the following:

Let $\mu$ be a Lebesgue-Stieltjes measure, and $f \in L^p(\mathbb{R}, \mathcal{B}(\mathbb{R}), \mu)$ with $0<p<\infty$.

Define $B_n=\{x: |x|\le n, |f(x)|\le n\}$ and $f_n=fI_{B_n}$ ($I$ is the indicator function). By the DCT we have, for every $\epsilon > 0$, there exists an $N_\epsilon$ such that for all $n \ge N_\epsilon$, $\int \left| f(x) - f_n(x)\right|^p d\mu < \epsilon$.

How is the DTC applied here? Thanks.

Answer 1

Note that $|f(x)-f_n(x)| \le |f(x)|$ and $|f|^p$ is integrable.

Since $|f|^p$ is integrable, $|f(x)| $ is finite ae. and so $f_n(x) \to f(x)$ ae.

Hence $\int |f(x)-f_n(x)|^p d \mu \to 0$.