ThmDex – An index of mathematical definitions, results, and conjectures.
Isotonicity of signed basic integral
Formulation 0
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}
Then \begin{equation} \int_X f \, d \mu \leq \int_X g \, d \mu \end{equation}
Formulation 1
Let $M = (X, \mathcal{F}, \mu)$ be a D1158: Measure space such that
(i) $f, g : X \to [-\infty, \infty]$ are each an D1921: Absolutely integrable function on $M$
(ii) \begin{equation} f \overset{a.e.}{\leq} g \end{equation}
Then \begin{equation} \int_X f \, d \mu \leq \int_X g \, d \mu \end{equation}
Proofs
Proof 0
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
(i) R1499: Real-linearity of signed basic integral
(ii) R1904: Isotonicity of finite real summation

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} $\square$