Let $F:[0,\infty)\rightarrow[0,\infty)$ be a continuous function, $X$ and $Y$ two non-negative random variables on some probability space $(\Omega,\mathcal{F},\mathbb{P})$, and
$$\mathbb{E}\big[\; F(X+Y) \;\big]<\infty$$
I am interested in the function
$$H(y)=\mathbb{E}\big[\; F(X+y) \;\big]$$
where $y\in[0,\infty)$. Is this function continuous? Is it continuous on some set $A\subset[0,\infty)$?
Intuitively, the random variable $F(X + y)$ is bounded on some (compact) set given by the random variable $Y$ and this should work.
If I understand corretly, those kind of continuity questions are usually answered via the dominated or monotone convergence theorem for Lebesgue integrals.
For example: given $y\in[0,\infty)$, we may take $y_{n}\rightarrow y$ and set $Z_{n}=F(X+y_{n})$, such that $Z_{n}\rightarrow Z=F(X+y)$. For dominated convergence however, we would need the $Z_{n}$ to be bounded by some integrable random variable.
Can anybody point me in the right direction or get me started?
Your help is highly appreciated! Thank you.