Bounded Measurable Functions on a Finite Measure Space Are Integrable

In the setting of Measurable Functions and the Lebesgue Integral: Notation, let $(X,\mathcal{F},\mu)$ be a measure space whose measure $\mu$ is finite, that is, $\mu(X)<\infty$. Let $f:X\to\mathbb{R}$ be measurable, and let $B\in\mathbb{R}$ with $|f(x)|\le B$ for every $x\in X$. Write $f+B:X\to\mathbb{R}$ for the pointwise sum of $f$ and the constant function with value $B$, so that

$$(f+B)(x)=f(x)+B\qquad(x\in X).$$

(i) Integrability. $\mu(X)$ is a real number, and $f\in\mathcal{L}^{1}(\mu)$.

(ii) The shifted function. The function $f+B$ lies in $\mathcal{M}^{+}$ and in $\mathcal{L}^{1}(\mu)$, its integral as an element of $\mathcal{M}^{+}$ equals its integral as an element of $\mathcal{L}^{1}(\mu)$, and

$$\int_{X}(f+B)\,d\mu=\int_{X}f\,d\mu+B\,\mu(X).$$