The \(L^p\) spaces package integrability into normed spaces. Their inequalities control approximation, convergence, and uniform integrability.
Lp spaces and norms
\(L^p\) identifies random variables that agree almost surely.
Definition. For \(1\le p\lt\infty\), \(L^p\) consists of random variables with \(\|X\|_p=(\mathbb E|X|^p)^{1/p}\lt\infty\), modulo almost-sure equality. \(L^\infty\) consists of essentially bounded variables with essential supremum norm.
H?lder, Jensen, and Minkowski
The basic inequalities make \(L^p\) usable as analytic spaces.
Theorem. If \(p,q\gt1\) and \(1/p+1/q=1\), then \(\mathbb E|XY|\le\|X\|_p\|Y\|_q\). Consequently, \(\|X+Y\|_p\le\|X\|_p+\|Y\|_p\) for \(p\ge1\). If \(\phi\) is convex, \(\phi(\mathbb E X)\le\mathbb E\phi(X)\) whenever both sides are meaningful.
Proof. H?lder follows from Young's inequality \(ab\le a^p/p+b^q/q\) after normalizing by \(\|X\|_p\) and \(\|Y\|_q\). Minkowski follows by applying H?lder to \(|X+Y|^p\le |X||X+Y|^{p-1}+|Y||X+Y|^{p-1}\). Jensen follows first for finite convex combinations and then by approximation or supporting-line arguments.
Corollary. On a probability space, if \(0<p<q\le\infty\), then \[ \|X\|_p\le\|X\|_q. \] For \(X,Y\in L^2\), \[ |\operatorname{Cov}(X,Y)| \le\sqrt{\operatorname{Var}(X)\operatorname{Var}(Y)}. \]
Proof. Jensen applied to the concave map \(u\mapsto u^{p/q}\) gives the norm comparison. Cauchy-Schwarz applied to \(X-\mathbb EX\) and \(Y-\mathbb EY\) gives the covariance inequality.
Example (equality in the covariance bound). Equality holds precisely when the centered variables are linearly dependent in \(L^2\): \[ Y-\mathbb EY=c(X-\mathbb EX)\quad\text{a.s.} \] Thus correlation \(\pm1\) is an almost-sure affine relation, not merely a strong informal association.
Uniform integrability
Uniform integrability is the missing hypothesis that upgrades convergence in probability to \(L^1\)-convergence.
Proposition. A family \(\mathcal X\subset L^1\) is uniformly integrable if \[ \lim_{K\to\infty}\sup_{X\in\mathcal X}\mathbb E[|X|1_{\{|X|\gt K\}}]=0. \] If \(X_n\to X\) in probability and \(\{X_n\}\) is uniformly integrable, then \(X\in L^1\) and \(X_n\to X\) in \(L^1\).
Proof. The family is uniformly bounded in \(L^1\). Every subsequence has an a.s.-convergent further subsequence; Fatou then gives \(X\in L^1\). Uniform integrability is equivalent, on a probability space, to uniform absolute continuity: \[ P(A)<\delta\quad\Longrightarrow\quad \sup_n\mathbb E[|X_n|\mathbf1_A]<\varepsilon. \] The family \(\{X_n-X\}\) inherits this property. Choose \(\eta>0\) with \(\eta<\varepsilon/2\). Since \(X_n\to X\) in probability, the event \(A_n=\{|X_n-X|>\eta\}\) has probability below the corresponding \(\delta\), eventually. Hence \[ \mathbb E|X_n-X| \le\eta+\mathbb E[|X_n-X|\mathbf1_{A_n}] <\varepsilon. \]
Proposition (moment criterion). If \[ \sup_{X\in\mathcal X}\mathbb E|X|^{1+\delta}<\infty \] for some \(\delta>0\), then \(\mathcal X\) is uniformly integrable.
Proof. On \(\{|X|>K\}\), \[ |X|\le K^{-\delta}|X|^{1+\delta}. \] Therefore \[ \sup_{X\in\mathcal X}\mathbb E[|X|\mathbf1_{\{|X|>K\}}] \le K^{-\delta} \sup_{X\in\mathcal X}\mathbb E|X|^{1+\delta} \longrightarrow0. \]
Example (bounded means are not enough). Let \[ X_n=n\mathbf1_{A_n},\qquad P(A_n)=1/n. \] Then \(\mathbb EX_n=1\) for every \(n\), but \[ \mathbb E[X_n\mathbf1_{\{X_n>K\}}]=1 \] whenever \(n>K\). Thus \(\{X_n\}\) is \(L^1\)-bounded but not uniformly integrable. Rare events can carry a fixed amount of expectation.