The Lebesgue Integral over ℝ of a Nonnegative Continuous Function as the Limit of Its Riemann Integrals over [−n, n]

In the settings of Measurable Functions and the Lebesgue Integral: Notation and Single-Variable Calculus on an Interval, assume the axiom of dependent choice.

(i) Compact intervals. Let $a,b\in\mathbb{R}$ with $a<b$, let $g:\mathbb{R}\to\mathbb{R}$ be a function such that $g|_{[a,b]}$ is continuous on $[a,b]$, and let $g\mathbf{1}_{[a,b]}:\mathbb{R}\to\mathbb{R}$ be the pointwise product of $g$ and the indicator function of $[a,b]$. Then $g\mathbf{1}_{[a,b]}\in\mathcal{L}^{1}(\lambda)$ and

$$\int_{\mathbb{R}}g\mathbf{1}_{[a,b]}\,d\lambda=\int_{a}^{b}g|_{[a,b]} .$$

Now let $h:\mathbb{R}\to\mathbb{R}$ be continuous on $\mathbb{R}$ with $h(x)\ge0$ for every $x\in\mathbb{R}$, so that $h\in\mathcal{M}^{+}(\mathbb{R},\mathcal{B}(\mathbb{R}))$, and for $n\in\mathbb{N}$ let

$$I_{n}=\int_{-n}^{n}h|_{[-n,n]} .$$

(ii) Exhaustion. For the integral of $h\in\mathcal{M}^{+}(\mathbb{R},\mathcal{B}(\mathbb{R}))$, in $[0,\infty]$,

$$\int_{\mathbb{R}}h\,d\lambda=\sup_{n\in\mathbb{N}}I_{n} .$$

(iii) Integrability. $h\in\mathcal{L}^{1}(\lambda)$ if and only if $\sup_{n\in\mathbb{N}}I_{n}<\infty$.

(iv) Limit. If $h\in\mathcal{L}^{1}(\lambda)$, then $I_{n}\to\int_{\mathbb{R}}h\,d\lambda$, the integral of $h\in\mathcal{L}^{1}(\lambda)$.