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• W1968173852 endingPage "705" @default.
• W1968173852 startingPage "703" @default.
• W1968173852 abstract "Let M 1 denote the set of Borel probability measures on the real line and M 1 the space of their Fourier transforms, or ‘characteristic functions’. It is well known that the natural correspondence between M 1 and M 1 is in fact a homeomorphism, provided M 1 is endowed with the usual weak topology relative to bounded continuous functions (which topology can be metrized by the construction of Lévy) and that M 1 is given the topology (also metric) appropriate to locally uniform convergence, which we call the ‘compact-open’ topology. In particular, if where μ n ∈ M 1 (and so φ n ∈ M 1 ), then φ n (λ)→ φ 0 (λ) uniformly on all compact sets if and only if μ n ⇒ μ 0 (weak convergence). However, the continuity theorem for characteristic functions implies that even if it is only known that φ n (λ)→ φ 0 pointwise (for all λ), the conclusion that μ n ⇒ μ 0 is still valid. This raises a question as to whether the pointwise (i.e. product) topology for M 1 might not, in fact, be equivalent to the compact-open one, since the corresponding notions of sequential convergence do coincide. The purpose of this note is to show that the answer is negative , even in a much more general setting." @default.
• W1968173852 created "2016-06-24" @default.
• W1968173852 creator A5017009905 @default.
• W1968173852 creator A5081425356 @default.
• W1968173852 date "1970-11-01" @default.
• W1968173852 modified "2023-10-05" @default.
• W1968173852 title "A remark on topologies for characteristic functions" @default.
• W1968173852 cites W4240370145 @default.
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• W1968173852 doi "https://doi.org/10.1017/s0305004100076702" @default.
• W1968173852 hasPublicationYear "1970" @default.
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