Why is convergence w.r.t $\mathcal L^p$-norm of a sequence $(f_n)$ of $\mathcal E-\mathcal B(\mathbb R)$-functions called "convergence in $p$-mean"?

Question

In measure-theory, why is convergence with respect to the $\mathcal L^p$-norm of a sequence $(f_n)_{n \in \mathbb N}$ of $\mathcal E-\mathcal B(\mathbb R)$-measurable functions called "convergence in $p$-mean" ?

I've tried reasoning about this name, but can't really find a good reason.

Answer 1

Let me guess:

If $(f_n)$ is a sequence as above which converges in $p$-norm to some $f$ then in functional analysis we write it as $$ \|f_n - f\|_p \to 0 \ \mbox{ as } n \to \infty,$$ however in probability theory expected value $\mathbb{E}$ (also called the mean) can be involved and we can write the above convergence as $$ \mathbb{E}(|f_n-f|^{p}) \to 0 \ \mbox{ as } n \to \infty.$$

I'm sorry if I'm wrong this would be my guess.

Answer 2

For finitely many numbers $a_1,\cdots,a_N\geq 0$, the $p$-mean is defined as $$\left(\frac{1}{N}\sum_{i=1}^N a_i^p\right)^{1/p}. $$ Assume wlog the domain has measure 1. Similarly with numbers, the $p$-mean of a function can be defined as $$\left(\int |f|^p\right)^{1/p},$$ and the convergence $f_n\to f$ in $L^p$ is equivalent to the convergence to $0$ of $p$-means of differences $|f_n-f|$, i.e. $$\left(\int |f_n-f|^p\right)^{1/p}\to0.$$