Exercise 1.4.36 (Linearity in measure)

Question

Let $(X,\mathcal{B},\mu)$ be a measure space, and let $f:X	o [0,+\infty]$ be measurable. Show that
\begin{enumerate}[label=(oman*)]
\item For every $c\in [0,+\infty]$ we have
$$\int_X f \ \mathrm{d}(c\mu) =  c	imes \int_{X} f \,\mathrm{d}{\mu}.$$
\item If $(\mu)_{n\in\mathbb{N}}$ is a sequence of measures on $(X,\mathcal{B})$ then
$$\int_X f \ \mathrm{d}\sum_{i=1}^{\infty}\mu_n = \sum_{i=1}^{\infty}\int_{X} f \,\mathrm{d}{\mu}_n.$$
\end{enumerate}

Elu Thingol · Accepted Answer

We have to demonstrate that
$$\sup\left\{\mathrm{Simp}\int_X h\ \mathrm{d}\left(\sum_{i=1}^{\infty}c\mu_night): \begin{array}{l} 	ext{$h$ is simple}\ h \leq f\end{array} ight\}
    = c	imes \sum_{i=1}^{\infty} \sup\left\{\mathrm{Simp}\int_{X} h \,\mathrm{d}\mu: \begin{array}{l} 	ext{$h$ is simple}\ h \leq f\end{array} ight\} $$
An arbitrary simple integral with respect to the linear combination of measures is contained in the set on the left-hand side
$$\mathrm{Simp}\int_X h\ \mathrm{d}\left(\sum_{i=1}^{\infty}c\mu_night)
    = \sum_{i=1}^{k}a_i \cdot \left(\sum_{i=1}^{\infty}c \mu_n(f^{-1}(a_i))ight)
    = c\cdot \sum_{i=1}^{\infty} \sum_{i=1}^{k}a_i \cdot \mu_n(f^{-1}(a_i))
    = c \cdot \sum_{i=1}^{\infty} \mathrm{Simp}\int_{X} h \,\mathrm{d}\mu_n$$
if and only if its product with $1/c$ and each summand is contained in the set on the right-hand side.

Exercise 1.4.36 (Linearity in measure)

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Exercise 1.4.36 (Linearity in measure)

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