Lebesgue Measure Space and a converging, increasing sequence. Show that the sequence in increasing.

Question

Let $(R,L,\lambda)$ be the Lebesgue measure space and let $f: R → R$ be a non-negative, measurable function. Define a sequence $f_n =f.\chi _{(\frac1n,1]} : R→R, n \in N$

(a) Show that the sequence is increasing and converges pointwise to a function $f\cdot\chi_{(0,1]\$

(b) Show that $$\int_{(0,1]}f d\lambda = \lim_{n \to \infty} \int_{(\frac1n,1]}fd\lambda.$$

For (a), I just showed that at $n=1$, the region is $(1,1]$, at $n=2$, it is $(\frac12,1)$, and continues to infinity where the region is $(0,1)$, showing that the region is increasing, and converging. However, I am not very sure of how to use this to solve part b. Any help would be appreciated.

Answer 1

• For (a) your approach works just fine. Alternatively, you could note that for $n>m$, $f_n-f_m=f\cdot\chi_{(\frac1n,\frac1m]}\ge 0$ and thus the sequence it's increasing. The pointwise convergence is then proven by $f-f_n=f\cdot \chi_{(0,\frac1n]}\to 0$.

• For (b), after having written the problem as

$$\int_{(0,1]}f(x)\text{d}\mu(x)=\lim_{n\to \infty}\int f_n(x)\text{d}\mu(x)$$

you can either

1. Use (a) and the monotone convergence theorem
2. Use the pointwise convergence and the fact that $f_n\le f$ to apply the dominated convergence theorem.