Let $(\mu_{n})$ sequence of probability measures of $\mathbb{R}^{d}$ converging to the prob measure $\mu$. Then by definition we know that $\int f d\mu_{n} \longrightarrow \int f d\mu $ for f continuous and bounded function.

I was wondering if I can write the formula above as $\int\int f(x-y) d\mu_{n}(x)d\mu_n(y) \longrightarrow \int\int f(x-y) d\mu(x)d\mu(y) $ ?

Or is the latter formula implied by the definition of weak convergence ?

Do I need additional assumptions on f ?