Rays, and Integrability and the Integral of a Real Function on a Borel Subset of the Real Line

Let $\mathbb{R}$ be the real numbers, $\mathcal{B}(\mathbb{R})$ the Borel $\sigma$-algebra and $\mathrm{Leb}$ Lebesgue measure on it.

1. (Rays)¶ For $a\in\mathbb{R}$, the open ray and the closed ray from $a$ are $(a,\infty)=\{t\in\mathbb{R}:\ a<t\}$ and $[a,\infty)=\{t\in\mathbb{R}:\ a\le t\}$; here $\infty$ is a symbol, not a number.

2. (Integral on a Borel set)¶ Let $B\in\mathcal{B}(\mathbb{R})$, let $f:B\to\mathbb{R}$, and let $\tilde{f}:\mathbb{R}\to\mathbb{R}$ be equal to $f$ on $B$ and to $0$ outside $B$. The function $f$ is integrable on $B$ if $\tilde{f}$ is measurable and integrable with respect to $\mathrm{Leb}$, and then its integral on $B$ is

$$\int_{B}f(t)\,dt=\int_{\mathbb{R}}\tilde{f}\,d\mathrm{Leb}.$$