The support-measure uncertainty inequality $\mathrm{\mid }E\mathrm{\mid }\mathrm{\mid }F\mathrm{\mid }\ge 1$

Given: Countable choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), a nonzero $f\in L^2({\mathbb{R}}^n;\mathbb{C})$, and Lebesgue measurable sets $E,F$ of finite measure with $f=0$ almost everywhere off $E$ and $\widehat{f}=0$ almost everywhere off $F$.

Countable choice is assumed; it is the hypothesis carried by the $L^1$ transform interface, the agreement theorem, and Plancherel below (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)).

For $g\in L^1({\mathbb{R}}^n;\mathbb{C})$ the $L^1$ transform is defined at every frequency, satisfies $\mathrm{\mid }\widehat{g}(\xi )\mathrm{\mid }\le \mathrm{\parallel }g{\mathrm{\parallel }}_1$, and is unchanged by null-set modifications of the representative (The integral transform is representative independent); it is continuous and vanishes at infinity (Riemann–Lebesgue lemma).

Hölder's inequality with conjugate exponents $p=q=2$: for measurable real $u\in {\mathcal{L}}^2$ and $v\in {\mathcal{L}}^2$ one has $\int \mathrm{\mid }uv\mathrm{\mid }\le \mathrm{\parallel }u{\mathrm{\parallel }}_2\mathrm{\parallel }v{\mathrm{\parallel }}_2$ (Holder's inequality for integrals, including the endpoint cases); $L^1$ and $L^2$ are the quotient spaces of The space $L^p(\mu )$ as the quotient by null functions with the norms of Complex Lp classes and Euclidean test-function conventions, and integrable functions that agree almost everywhere have equal integrals (Two integrable functions are equal almost everywhere exactly when all of their indefinite integrals agree).

If $g\in L^1\cap L^2$, its bounded continuous $L^1$ transform represents the Plancherel transform ${\mathcal{F}}_{2}g$ almost everywhere (Agreement of the integral and L2 transforms), and Plancherel gives $\mathrm{\parallel }{\mathcal{F}}_{2}g{\mathrm{\parallel }}_2=\mathrm{\parallel }g{\mathrm{\parallel }}_2$ (Plancherel theorem).