Let $(\Omega_{1},\mathcal{F}_{1},\mathbb{P}_{1})$ and $(\Omega_{2},\mathcal{F}_{2},\mathbb{P}_{2})$ be two probability measure spaces and let $f:\Omega_{1}\rightarrow \mathbb{R}$, $g:\Omega_{2}\rightarrow \mathbb{R}$ be two random variables.
Assume that there exists a measurable function $H:\mathcal{B}(\mathbb{R})\times\mathcal{B}(\mathbb{R})\rightarrow \mathbb{R}$ such that $$(\mathbb{P}_{1}\times\mathbb{P}_{2})(\{(x,y)\in \Omega_{1}\times\Omega_{2}:f(x)\in A, g(y)\in B\})=\iint_{A\times B}H(s,t)d(\mu\times\mu)(s,t),$$ for any Borel sets $A$ and $B$ in $\mathbb{R}$. Here $\mathcal{B}(\mathbb{R})$ denotes the Borel set and $\mu$ is the Borel measure on $\mathbb{R}$.
Question: How can we prove that for any $c\in\mathbb{R}$,
$$(\mathbb{P}_{1}\times\mathbb{P}_{2})(\{(x,y)\in \Omega_{1}\times\Omega_{2}:f(x)+g(y)<c\})=\int_{-\infty}^{\infty}\int_{-\infty}^{c-t}H(s,t)d(\mu\times\mu)(s,t).$$
As standard argument using Monotone Class Theorem shows that $$(\mathbb{P}_{1}\times\mathbb{P}_{2})(\{(x,y)\in \Omega_{1}\times\Omega_{2}:(f(x),g(y)) \in E\})=\iint_{E}H(s,t)d(\mu\times\mu)(s,t)$$ for any $E \in B ( \mathbb R) \times B ( \mathbb R)$ (the product sigma algbebra). Take $E=\{(s,t): s+t<c\}$ in this formula and note that $I_{\{s+t <c\}}(s,t)=1$ iff $I_{\{s: s<c-t\}}(s,t) =1$.