### Abstract

Let (Ω,?, p) be a measure space, µ(Ω) ∞. Let X be a sequence of measurable functions on Ω taking values in a compact metric space M. The set of bounded shopping times r for the X isa directed set under the obvious ordering. The following theorem is proved: X converges pointwise almost everywhere if and only if the generalized n sequence 4Ø(Xt)du converges for every continuous function Ø on M. The martingale theorem is proved as an application.

Original language | English (US) |
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Pages (from-to) | 395-398 |

Number of pages | 4 |

Journal | Proceedings of the American Mathematical Society |

Volume | 46 |

Issue number | 3 |

DOIs | |

State | Published - Dec 1974 |

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## Cite this

Baxter, J. R. (1974). Pointwise in terms of weak convergence.

*Proceedings of the American Mathematical Society*,*46*(3), 395-398. https://doi.org/10.1090/S0002-9939-1974-0380968-7