integration with respect to a sequence of measures can someone help me?
Let $(\mu_n)$ be a sequence of measures on $(X, \mathcal{A})$ and $\mu:= \sum_{n \geq 1} \mu_n$. Then, for every measurable $f: X \rightarrow [0,\infty]$,
$$\int_X f d\mu = \sum_{n \geq 1} \int_X f d\mu_n.$$

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*Let $A \in \mathcal{A}$ and $f := 1_A$. Then, by definition, $$\int f \, d\mu = \mu(A) = \sum_{n \geq 1} \mu_n(A) = \sum_{n \geq 1} \int f \, d\mu_n.$$ This shows that the claim holds for indicator functions.

*By linearity of the integral, the equality extends to all simple functions, i.e. for all $f$ of the form $$f = \sum_{j=1}^m c_j 1_{A_j}$$ we have $$\int f \, d\mu = \sum_{n \geq 1} \int f \, d\mu_n. \tag{1}$$

*Let $f \geq 0$ be measurable. Then there exists a sequence of simple functions $(f_k)_{k \in \mathbb{N}}$ such that $0 \leq f_k \uparrow f$. By the monotone convergence theorem, $$\int f \, d\mu = \sup_{k \in \mathbb{N}} \int f_k \, d\mu \stackrel{(1)}{=} \sup_{k \in \mathbb{N}} \sup_{N \in \mathbb{N}} \sum_{n=1}^N \int f_k \, d\mu_n.$$ Using that $$\sup_{k \in \mathbb{N}} \sup_{j \in \mathbb{N}} \beta_{jk} = \sup_{j \in \mathbb{N}} \sup_{k \in \mathbb{N}} \beta_{jk}$$ holds for any $\beta_{jk} \geq 0$, $j,k \in \mathbb{N}$, we get $$\int f \, d\mu = \sup_{N \in \mathbb{N}} \sup_{k \in \mathbb{N}} \sum_{n=1}^N \int f_k \, d\mu_n = \sup_{N \in \mathbb{N}} \sum_{n=1}^N \int f \, d\mu_n = \sum_{n=1}^{\infty} \int f \, d\mu_n.$$

