Convergence in Lp ($X_n\to X\text{ in }{L}^p$)

Random variables converge in $L^p$ when $\mathbb{E}|X_n-X{|}^p\to 0$.

Convergence in L1 means that the expected value of the absolute error tends to zero. It implies convergence in probability and convergence of expected values.

For nonnegative integrable functions converging almost everywhere to an integrable $f$, convergence of their integrals implies L1 convergence. Indeed $\min (f_n,f)\to f$ almost everywhere and is dominated by $f$, so the dominated convergence theorem gives convergence of its integral. Use $|f_n-f|=f_n+f-2\min (f_n,f)$. This applies to probability densities, giving convergence in total variation distance.

Convergence in $L^2$ means $\mathbb{E}[(X_n-X{)}^2]\to 0$. It implies convergence in probability by Markov inequality.

If square-integrable random variables satisfy $X_n\to X$ almost surely, then $X_n\to X$ in $L^2$ exactly when $\mathbb{E}[X_n^2]\to \mathbb{E}[X^2]$. Boundedness in $L^2$ and almost-sure convergence first give weak convergence in the Hilbert space $L^2$; weak convergence together with convergence of norms gives strong convergence.