## Abstract

Being motivated by Dini’s Theorem, the uniform convergence of a sequence of functions is characterized by that of another decreasing sequence of functions. This result allows us to describe the uniform convergence of a sequence of measurable functions on a closed interval in terms of that of a decreasing sequence of Lebesgue integrable functions. For a sequence of continuous functions on a compact set, its uniform convergence is further characterized by the pointwise convergence of another decreasing sequence of continuous functions. It also turns out that the uniform convergence of a sequence of Riemann integrable functions on a closed interval can be determined by that of another decreasing sequence of Riemann integrable functions. As applications, characterizations for a uniformly convergent series of functions have been presented and one fixed point theorem has been established.

Original language | English |
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Pages (from-to) | 27-36 |

Number of pages | 10 |

Journal | Journal of Mathematical Analysis |

Volume | 11 |

Issue number | 6 |

Publication status | Published - 2020 |

## Keywords

- Riemann integrable function
- Uniform convergence
- fixed point theorem
- measurable function
- pointwise convergence