Integrals of Uniform Limits and of Uniformly Convergent Series

In the setting of The Real Line: Standing Notation and Background for Calculus on Intervals, let $a,b\in\mathbb{R}$ with $a<b$. A map $h:[a,b]\to\mathbb{R}$ is bounded if there is $B\in\mathbb{R}$ with $|h(t)|\le B$ for every $t\in[a,b]$; integrability on $[a,b]$ and the integral $\int_{a}^{b}h$ are as in The Riemann Integral on a Closed Interval via Lower and Upper Integrals (iv) and The Riemann Integral on a Closed Interval via Lower and Upper Integrals (v). $\mathbb{R}^{[a,b]}$ is the set of maps from $[a,b]$ to $\mathbb{R}$ of Sets and Maps: Ordinary Notation (v). Uniform convergence on $[a,b]$ is as in Pointwise and Uniform Convergence of a Sequence of Real-Valued Maps (ii), with $D=S=E=[a,b]$; convergence of a sequence in $\mathbb{R}$ is as in Convergent Sequences of Real Numbers (i), and convergence and the sum of a series as in Series of Real Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence (ii).

(i) Let $(f_{k})_{k\in\mathbb{N}}$ be a sequence in $\mathbb{R}^{[a,b]}$ such that $f_{k}$ is bounded and integrable on $[a,b]$ for every $k\in\mathbb{N}$, and let $f:[a,b]\to\mathbb{R}$ be a map such that $(f_{k})_{k\in\mathbb{N}}$ converges uniformly to $f$ on $[a,b]$. Then $f$ is bounded and integrable on $[a,b]$, and the sequence $\bigl(\int_{a}^{b}f_{k}\bigr)_{k\in\mathbb{N}}$ converges to $\int_{a}^{b}f$.

(ii) Let $(g_{k})_{k\in\mathbb{N}}$ be a sequence in $\mathbb{R}^{[a,b]}$ such that $g_{k}$ is bounded and integrable on $[a,b]$ for every $k\in\mathbb{N}$; for $n\in\mathbb{N}$ let $s_{n}:[a,b]\to\mathbb{R}$ be the map $t\mapsto\sum_{k=1}^{n}g_{k}(t)$; and let $s:[a,b]\to\mathbb{R}$ be a map such that $(s_{n})_{n\in\mathbb{N}}$ converges uniformly to $s$ on $[a,b]$. Then $s$ is bounded and integrable on $[a,b]$, the series $\sum_{k=1}^{\infty}\int_{a}^{b}g_{k}$ converges, and

$$\sum_{k=1}^{\infty}\int_{a}^{b}g_{k}=\int_{a}^{b}s .$$