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Singular strictly increasing functions and a problem on partitions of closed intervals
Authors:I S Kats
Institution:(1) Odessa National Academy of Food Technologies, Russia
Abstract:We establish that the problem of constructing a strictly increasing singular function is equivalent to the problem of constructing subsets ></img>                                </span>                              </span> and <span class= ></img>                                </span>                              </span> of a closed interval <em>a; b</em>] ? ? such that (1) <span class= ></img>                                </span>                              </span> ∩ <span class= ></img>                                </span>                              </span> = ø; (2) <span class= ></img>                                </span>                              </span> ∪ <span class= ></img>                                </span>                              </span> = <em>a; b</em>]; (3) the Lebesgue measures of the intersections of <span class= ></img>                                </span>                              </span> and <span class= ></img>                                </span>                              </span> with an arbitrary interval <em>J</em> ? <em>a; b</em>] are positive.</td>
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Keywords:singular function  Cantor set  perfect set  heavily intermittent partition  Borel set  Lebesgue measurable set  completely additive function
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