Uniform Convergence and Integration

Theorem (Uniform Convergence and Integration)

Let $E=[a,b]$ and $\alpha :[a,b]\to \mathbb{R}$ be increasing. Let $f_n\in \mathcal{C}(B)$ such that $f_n\in \mathcal{R}(\alpha )$. If $f_n\rightrightarrows f$ then $f\in \mathcal{R}(\alpha )$ and

$${\int }_a^{b}fd\alpha =\underset{n\to \infty }{\lim }{\int }_a^b{f}_{n}d\alpha =0$$

Proof: We see that $\forall \epsilon >0,\exists N\in \mathbb{N}$ such that $|f_n-f|<\epsilon$ for all $x\in [a,b]$. So,

$$f_n(x)-\epsilon \le f(x)\le f_n(x)+\epsilon \forall x\in [a,b]$$

Then we can compare the LHS and the RHS by

$${\int }_a^b(f_n(x)-\epsilon )d\alpha \le \underline{{\int }_a^b}\le \overline{{\int }_a^b}\le {\int }_a^b(f_n(x)+\epsilon )d\alpha$$

such that RHS - LHS = $2\epsilon (\alpha (b)-\alpha (a))$.

Related, Riemann-Stieltjes Integral, Pointwise and Uniform Convergence.