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 $fcdotchi_{(0,1]$
(b) Show that $$int_{(0,1]}f dlambda = lim_{n to infty} int_{(frac1n,1]}fdlambda.$$
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.

Caffeine · Answer

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

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

you can either

Use (a) and the monotone convergence theorem
Use the pointwise convergence and the fact that $f_nle f$ to apply the dominated convergence theorem.

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

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