## Differentiation of Integrals in Rn |

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### Contents

1 | |

2 | |

Covering theorems of the Whitney type | 9 |

Covering theorems of the Vitali type tº e º º º s is e º º º e º ºs | 19 |

Weak type 11 of the maximal operator | 36 |

Differentiation bases and the maximal operator associated to them | 42 |

The rotation method in the study of the maximal operator | 51 |

The space L1 + log L Integrability properties of the maximal oper | 60 |

The basis 8 3 is not a density basis | 115 |

CHAPTER | 134 |

Bases of unbounded sets and starshaped sets º e s s e 1141 | 147 |

The theorem of de Possel + + m + 4 + 1148 | 153 |

A problem related to the interval basis | 165 |

ON THE HALO PROBLEM | 177 |

An application of the extrapolation method of Yano | 183 |

APPENDIX I | 190 |

Density bases Theorems of BusemannFeller | 66 |

Individual differentiation properties | 77 |

THE INTERVAL BASIS | 92 |

CHAPTER V | 109 |

by Roberto Moriyên Žſ | 211 |

221 | |

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### Common terms and phrases

Assume B-F basis bounded measurable set bounded set characteristic function closed cubic intervals concludes the proof constant convex sets covering property covering theorem cubes cubic intervals centered define density basis density property differentiates ſf differentiation basis differentiation properties differentiation theory disjoint sequence Euclidean balls exists F basis f e L*(R f e Lice finite fixed function f Guzmán halo function Hence homothetic interval Q(x invariant by homothecies Lebesgue measure lemma Let f e maximal operator associated measurable function metric space Mf(x Nikodym null set obtain open bounded open cubic interval open intervals open set order to prove points problem proof of Theorem rectangles Remark ſ f satisfies sequence Q side length ſº Theorem 1.1 theorem of Besicovitch theorem of Vitali triangles Vitali covering weak type 1,1 Zygmund