Let $M = (X, \mathcal{F}, \mu)$ be a
D1158: Measure space such that
| (i) |
$f, g : X \to [-\infty, \infty]$ are each a D5566: Measurable basic function on $M$
|
| (ii) |
\begin{equation}
\Vert f \Vert_{L^1}, \Vert g \Vert_{L^1}
< \infty
\end{equation}
|
| (iii) |
\begin{equation}
f
\overset{a.e.}{\leq} g
\end{equation}
|
Result
R1022: Partition of basic function into positive and negative parts provides the partitions $f = f^+ - f^-$ and $g = g^+ - g^-$. Since $f \leq g$ almost everywhere, then also $f^+ \leq g^+$ almost everywhere, whence
R1514: Isotonicity of signed basic integral implies
\begin{equation}
\int_X f^+ \, d \mu
\leq \int_X g^+ \, d \mu
\end{equation}
Similarly, we have $f^- \geq g^-$ almost everywhere and thus $\int_X f^- \, d \mu \geq \int_X g^- \, d \mu$. Negating both sides, this gives
\begin{equation}
- \int_X f^- \, d \mu \leq - \int_X g^- \, d \mu
\end{equation}
Applying results
and the above two inequalities, we may then conclude
\begin{equation}
\int_X f \, d \mu = \int_X f^+ \, d \mu - \int_X f^- \, d \mu \leq \int_X g^+ \, d \mu - \int_X g^- \, d \mu = \int_X g \, d \mu
\end{equation}
$\square$
