Definition 3

Tag 02I1, permanent. Cite it as [Bourbaki, Tag 02I1].

Théories spectrales, chapter II, § 1. Transformation de Fourier

Let $\mu\in \mathscr{M}^1(G)$ be a bounded complex measure on G. The Fourier transform of $\mu$ is called the function $\mathscr{F}_G(\mu)$ on $\widehat{G}$ defined by

\mathscr{F}_G(\mu)(\widehat{x}) =\int_G\overline{\langle\widehat{x}, x\rangle}d\mu(x) \tag{5}

The Fourier cotransform of $\mu$ is called the function $\overline{\mathscr{F}}_G(\mu)$ on $\widehat{G}$ defined by

\overline{\mathscr{F}}_G(\mu)(\widehat{x}) =\int_G\langle \widehat{x}, x\rangle d\mu(x) \tag{6}

When there is no ambiguity concerning the group G considered, one shall also write $\mathscr{F}(\mu)$ and $\overline{\mathscr{F}}(\mu)$ . One also sometimes denotes $\widehat{\mu}=\mathscr{F}_G(\mu)$ .

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

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