Let $L^\infty(Ω,F,P)$ be the vector space of bounded random variables $(X ∈ L^\infty (Ω,F,P)$ means that there exists a constant C such that $|X(ω)|≤C$, a.s.$)$. Show that $$L^\infty(Ω,F,P)⊂L^2(Ω,F,P)⊂L^1(Ω,F,P)$$
2026-04-28 10:52:18.1777373538
Embedded Lp spaces
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It is a consequence of Holder inequality $$ E[|XY|]\leq E[|X|^p]^{1/p}E[|Y|^q]^{1/q} $$