Nonnegative Combinations of Finite Measures and Their Integrals

In the setting of Radon Measures on Locally Compact Metric Spaces: Standing Notation, let $(X,\mathcal{F})$ be a measurable space, let $\mu_{1}$ and $\mu_{2}$ be finite measures on $(X,\mathcal{F})$, and let $a_{1},a_{2}\in\mathbb{R}$ with $a_{1},a_{2}\ge0$. Let $a_{1}\mu_{1}+a_{2}\mu_{2}$ be the function on $\mathcal{F}$ with

$$(a_{1}\mu_{1}+a_{2}\mu_{2})(A)=a_{1}\mu_{1}(A)+a_{2}\mu_{2}(A)\qquad(A\in\mathcal{F}).$$

(i) A finite measure. $a_{1}\mu_{1}+a_{2}\mu_{2}$ is a finite measure on $(X,\mathcal{F})$.

(ii) Integrals. For every $g\in\mathcal{M}^{+}$,

$$\int g\,d(a_{1}\mu_{1}+a_{2}\mu_{2})=a_{1}\int g\,d\mu_{1}+a_{2}\int g\,d\mu_{2},$$

with the sum and the products formed in $[0,\infty]$. Every $f\in\mathcal{L}^{1}(\mu_{1})\cap\mathcal{L}^{1}(\mu_{2})$ lies in $\mathcal{L}^{1}(a_{1}\mu_{1}+a_{2}\mu_{2})$, and

$$\int f\,d(a_{1}\mu_{1}+a_{2}\mu_{2})=a_{1}\int f\,d\mu_{1}+a_{2}\int f\,d\mu_{2}.$$