# Infinite sum of right-continuous and monotone non-decreasing functions and its Lebesgue-Stieltjes integral

Let $$g_n: [0,\infty) \to [0,\infty)$$ be a right-continuous and monotone non-decreasing function for every $$n \in \mathbb N$$. Define the pointwise limit $$g(x) = \sum_n g_n(x)$$ and assume that it is finite for all $$x \geq 0$$. Then it is true that:

1. $$g$$ is also right-continuous and monotone non-decreasing.

2. For every $$f: [0,\infty) \to \mathbb R$$ with existing Lebesgue-Stieltjes integral $$\int f \mathrm d g$$ we have that $$\int f \mathrm d g = \sum_{n \in \mathbb N} \int f \mathrm d g_n$$

It's been quite a while since I've taken measure and integration theory and the Lebesgue-Stieltjes integral is new to me.

For 1.: The monotony is clear by induction for finite sums. I'm a little embarrassed to say that I have difficulty coming up with a rigorous argument for the infinite sum although it seems obvious that $$\sum_n a_n \leq \sum_n b_n$$ where $$a_n \leq b_n$$ for all $$n$$. Concerning right-continuity, it is known that the infinite sum of continuous functions need not be continuous in general. But my Analysis is not strong enough to figure this out for this case.

So I tried to resort to measure theory: For every $$f_n$$ there exists a unique measure $$\mu_n$$ by setting $$\mu_n([0,x])=f_n(x)$$ for all $$x \geq 0$$ and applying the uniqueness theorem for measures, since the intervals $$[0,x]$$ generate the Borel-$$\sigma$$-algebra and are closed under intersection of sets. Now, the infinite sum $$\sum_n \mu_n$$ of measures should again be a measure $$\mu$$. By definition we have $$g(x)=\mu([0,x])$$ for all $$x\geq 0$$. If $$\mu$$ is in fact a measure, then it should follow directly that $$g$$ has the required properties, right?

However, this would technically require to show the lemmata that the countable sum of measures is a measure and that the correspondence between a non-negative, right-continuous and monotone non-decreasing function and a measure is one-to-one. I would like to be able to verify the statement by more analytic means.

Another question that pops into my head is whether the finiteness of $$g$$ is necessary for 1. to hold. (I see that it is necessary for 2.)

For 2.: Again, this corresponds to the statement that $$\int f \mathrm d \sum_n \mu_n = \sum_n \int f \mathrm d \mu_n.$$

I faintly remember having seen a statement like this. My first guess would be that this can be shown by measure theoretic induction: First for indicator functions, then for simple functions and so on. Is there any cuter way to do this?