The Stieltjes interval set function is finitely additive on the half-open interval algebra

Collect every finite endpoint appearing among the $I_i$ and $J_j$, and adjoin $-\mathrm{\infty }$ or $+\mathrm{\infty }$ when a left or right ray occurs. This gives an increasing list $p_0<\cdots <p_r$ such that each $I_i$ and each $J_j$ is a disjoint union of consecutive h-cells $C_{\mathrm{\ell }}:=(p_{\mathrm{\ell }-1},p_{\mathrm{\ell }}]$, with the first or last cell possibly a ray.

In the unbounded cases this is exactly the truncation/supremum convention built into The interval set function attached to a nondecreasing right-continuous function. Thus each decomposition gives the same total, namely the sum of the cell values over those $C_{\mathrm{\ell }}$ contained in $E$. So ${\mu }_{0,F}(E)$ is well defined. [step 1.1, L1, algebra]

Refine them to a common endpoint grid as in step 1.1, and sum over the grid cells. The cells belonging to $E\cup G$ are exactly the disjoint union of the cells belonging to $E$ and the cells belonging to $G$, so the corresponding cell sums add:

${\mu }_{0,F}(E\cup G)={\mu }_{0,F}(E)+{\mu }_{0,F}(G)\mathrm{.}$

Together with step 2.1 this proves the claim. [step 2.1, given, algebra] ∎