(PTE) 1.6. $L^p$ space

$\newcommand{\argmin}{\mathop{\mathrm{argmin}}\limits}$ $\newcommand{\argmax}{\mathop{\mathrm{argmax}}\limits}$

In the previous section, we defined Lebesgue integrability and write $f \in L^1$ for such function $f$. We also defined the $L^p$-norm. $L$ in these notations stands for “Lebesgue” and $L^p$ for $p \ge 1$ becomes a space of functions that is integrable in the order of $p$. We name it $L^p$ space.


$L^p$ space on probability space


$L^p = L^p(\Omega,\mathcal{F},P) := \{X: E|X|^p < \infty\}.$
$\|X\|_p := (E|X|^p)^{1/p},~ X\in L^p$ is the $L^p$-norm.

$L^p$-norms are monotone. That is, $|X|_p \le |X|_q,~ 1 \le p \le q$. Thus if $X \in L^q$, then $X \in L^p$ for $1 \le p < q$.

Convergence in $L^p$

Let $X_n, X$ be random variables and $X_n \in L^p,~ \forall n$. If $X \in L^p$ and $\|X_n - X\|_p \to 0$, then we say $X_n$ converges to $X$ in $L^p$ and write $X_n \overset{L^p}{\to} X$.

Or we could just say $X_n \to X$ in $L^p$. It is known that for $p \ge 1$, $L^p$ are closed (Banach) spaces. If $p=2$, it becomes a Hilbert space since we can define the inner product as $\langle X, Y \rangle = EXY$.



Acknowledgement

This post series is based on the textbook Probability: Theory and Examples, 5th edition (Durrett, 2019) and the lecture at Seoul National University, Republic of Korea (instructor: Prof. Johan Lim).