In mathematics, Bochner's theorem (named for Salomon Bochner) characterizes the Fourier transform of a positive finite Borel measure on the real line. More generally in harmonic analysis, Bochner's theorem asserts that under Fourier transform a continuous positive-definite function on a locally compact abelian group corresponds to a finite positive measure on the Pontryagin dual group. The case of sequences was first established by Gustav Herglotz (see also the related Herglotz representation theorem.) The theorem for locally compact abelian groups Bochner's theorem for a locally compact abelian group G, with dual group , says the following: Theorem For any normalized continuous positive-definite function f on G (normalization here means that f is 1 at the unit of G), there exists a unique probability measure μ on such that i.e. f is the Fourier transform of a unique probability measure μ on . Conversely, the Fourier transform of a probability measure on is necessarily a normalized continuous positive-definite function f on G. This is in fact a one-to-one correspondence. The Gelfand–Fourier transform is an isomorphism between the group C*-algebra C*(G) and C0(Ĝ). The theorem is essentially the dual statement for states of the two abelian C*-algebras. The proof of the theorem passes through vector states on strongly continuous unitary representations of G (the proof in fact shows that every normalized continuous positive-definite function must be of this form). Given a normalized continuous positive-definite function f on G, one can construct a strongly continuous unitary representation of G in a natural way: Let F0(G) be the family of complex-valued functions on G with finite support, i.e. h(g) = 0 for all but finitely many g. The positive-definite kernel K(g1, g2) = f(g1 − g2) induces a (possibly degenerate) inner product on F0(G). Quotiening out degeneracy and taking the completion gives a Hilbert space whose typical element is an equivalence class [h]. For a fixed g in G, the "shift operator" Ug defined by (Ug)(h) (g') = h(g − g), for a representative of [h], is unitary. So the map is a unitary representations of G on . By continuity of f, it is weakly continuous, therefore strongly continuous. By construction, we have where [e] is the class of the function that is 1 on the identity of G and zero elsewhere. But by Gelfand–Fourier isomorphism, the vector state on C*(G) is the pull-back of a state on , which is necessarily integration against