Definition (Convergence in Expectation)

Let $(\Omega,\mathcal{F})$ be a Measurable Space. Let $(X_{n})_{n\in\mathbb{N}}$ be a sequence of rvs with densities $(f_{n})_{n\in\mathbb{N}}$ and let $X$ be another rv with density $f$. We say $X_{n}$ converges to $X$ in mean to the order $p$ if they converge in Lp: $$\lVert f_{n}-f \rVert_{p} \to0 ,\,1\le p<\infty$$or $$\mathbb{E}[|f_{n}-f|^{p}]=\int\limits _{\mathbb{R}}\left| f_{n}(x)-f(x) \right|^{p} \, \mu(dx) \to 0$$written as $$X_{n}\xrightarrow{L^{p}}X.$$