By Leadbetter R., Cambanis S., Pipiras V.

ISBN-10: 1107020409

ISBN-13: 9781107020405

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**Example text**

The result which we shall ﬁnd most useful is based on the following lemma. 4 Let E be a nonempty class of sets which is closed under the formation of intersections (E ∩ F ∈ E whenever E, F ∈ E). Then D = D(E) is also closed under the formation of intersections. Proof For any set E let DE = {F : F ∩ E ∈ D(E)}. Clearly if F ∈ DE then E ∈ DF . Now for a given ﬁxed E, DE is a D-class. (For if F, G ∈ DE and F ⊃ G, then (F –G)∩E = (F ∩E)–(G∩E) which is the proper difference of two sets of D(E) and hence belongs to D(E) so that F – G ∈ DE .

Show that μ is continuous from above at ∅ but is not a measure. 6? 9 Let X be any space with two or more points. Write μ(∅) = 0 and μ(E) = 1 for E ∅. Is μ an outer measure, a measure? 10 If μ* is an outer measure and E, F are two sets, E being μ* -measurable, show that μ* (E) + μ* (F) = μ* (E ∪ F) + μ* (E ∩ F). 11 Let x0 be a ﬁxed point of space X. Is μ* (E) = χE (x0 ) an outer measure? 12 Let R be a ring of subsets of a countable set X with the property that every nonempty set in R is inﬁnite and such that S(R) is the class of all subsets of X (give an example of such on (X, R)).

3. 13 If X is any nonempty set, show that the class P consisting of ∅ and all onepoint sets is a semiring. Is it a ring? A ﬁeld? 14 Show that if E is a nonempty class of sets, then every set in S(E) can be covered by a countable union of sets in E. 15 Let E be a class of sets. Is there a smallest semiring P(E) containing E? 16 Show the “monotone class theorem”, viz. the monotone class M(R) generated by a ring R is the same as the σ-ring S(R) generated by R. 4, so that M(R) is a ring. 17 Show that a nonempty class which is closed under the formation of intersections, proper differences and countable disjoint unions, is a σ-ring.

### A Basic Course in Measure and Probability: Theory for Applications by Leadbetter R., Cambanis S., Pipiras V.

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