A function $\phi:\mathbb{R}^d\to\mathbb{R}$ is lower semi-continuous if $(x_n)_n\to x\Rightarrow \liminf\limits_n\phi(x_n)\geqslant \phi(x)$ or (equivalent) $\forall a\in\mathbb{R},\{\phi\leqslant a\}$ is closed.
Show that the convolution of two nonnegative measurable functions is upper semi-continuous. Any hints ? Thanks !
My work:
I wrote $$\int_{\mathbb{R^d}}f(x_n-y)g(y)dy$$ but only having nonnegativeness does not tell me anything on this integral, especially how this sequence would converge...