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For a Schwartz function on the real numbers, its periodization is a smooth periodic function of period one. Rapid decay gives uniform convergence on $[0,1]$ of the series and each differentiated series. Under the Fourier transform convention $\hat{f}(\xi )=\int f(x)e^{-2\pi ix\xi }dx$, its $m$th Fourier coefficient is $\hat{f}(m)$: integrate termwise over one period and combine the translated intervals. The rapidly decreasing coefficients give a Fourier series with uniform convergence. Evaluation at zero proves the Poisson summation formula.

The indicator of the rational numbers in the real numbers is discontinuous everywhere because both the rationals and irrationals are dense. The discontinuity theorem for Baire class one functions therefore shows that it is not Baire class one.

A real vector space is a vector space over the field $\mathbb{R}$ of real numbers.

An uncountable subset $S$ of the real numbers whose intersection with every null set for Lebesgue measure is countable. Testing only Borel sets suffices because every Lebesgue-null set has a Borel null superset. No measurability of $S$ is assumed.

The Skorokhod embedding theorem represents a centered probability distribution $\mu$ on the real numbers with finite variance as $B_T\sim \mu$ for a stopping time $T$ of Brownian motion. One may choose $T$ so that the stopped process $(B_{t\wedge T})$ is uniformly integrable and $\mathbb{E}T=\int x^2\mu (dx)$.

A square root of an element $x$ is an element $y$ such that $y^2=x$. The notation $\sqrt{x}$ denotes the nonnegative square root when $x$ is a nonnegative real number.