Monotone convergence theorem to evaluate improper integral

Question

My book says that the equality $$\int_{(0,1]}x^{-3/4}d\mu=\lim_{t\rightarrow 0^+}\int_{[t,1]}x^{-3/4}d\mu$$ follows from the monotone convergence theorem. Why is it so? I can't see how to apply monotone convergence theorem here.

Answer 1

Hint: For each $k\in\mathbb N^*$, define $f_k(x)$ on $(0,1]$ by $x^{-3/4}$ if $x\geq \frac{1}{k}$ and 0 otherwise. This is an pointwise increasing sequence of measurable functions (assuming $\mu$ is the Lebesgue measure), can you find its limit?

Answer 2

You take the sequence of functions $f_n = x^{-3/4}$ for $1 \geq x \geq 1/n$, and $0$ otherwise. Then you see that this sequence of functions are monotonically increasing to $ x^{-3/4}$ defined on $0 \leq x \leq 1$.