Measure, Integral, Derivative by Sergei Ovchinnikov

Author:Sergei Ovchinnikov
Language: eng
Format: epub
Publisher: Springer New York, New York, NY

(Carathéodory Criterion).

2.23 Show that for any bounded set E, the following statements are equivalent: (a) E is measurable.

(b)Given any ɛ > 0, there is an open set G ⊇ E such that

(c)Given any ɛ > 0, there is a closed set F ⊆ E such that

2.24 Show that a set E is measurable if and only if for each ɛ > 0, there is a closed set F and open set G for which F ⊆ E ⊆ G and m  ∗ (G ∖ F) < ɛ.

2.25 Let E be a measurable set and ɛ > 0. Show that E is a union of a finite family of pairwise disjoint measurable sets, each of which has measure at most ɛ.

2.26 Let E be a measurable set. Show that for each ɛ > 0 there is a finite family of pairwise disjoint open intervals {I i } i ∈ J such that

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