By John G. Kemeny, J. Laurie Snell, Anthony W. Knapp

ISBN-10: 0387901779

ISBN-13: 9780387901770

ISBN-10: 3540901779

ISBN-13: 9783540901778

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**Sample text**

Finally we deﬁne is ∞ Eab . Ea = b=a By the basic convexity property we have b P r(Eab ) ≤ P r({s : | m=a S m2 1 − | > }). 2 m 2 Now using Chebyshev’s inequality just as we did in the proof of the weak law we get P r(Eab ) b 1 . 2 m m=a 1 ≤ 2 4 As we let b go to inﬁnity and use the fact that for ﬁxed a the family Eab is a nested increasing family of sets, so that by continuity we get 1 P r(Ea ) = lim P r(Eab ) ≤ 2 b− →∞ 4 ∞ 1 . m2 m=a What does it mean for a sequence to be in a set of the form Ea ?

Applied to our example it produces the Lebesgue measure. 7 Let X be a set and let µ∗ be an outer measure deﬁned on X. Denote by Σ the collection of all subsets, say A, such that for every subset E of X we have µ∗ (E) = µ∗ (A ∩ E) + µ∗ (Ac ∩ E). For all A in Σ deﬁne µ(A) = µ∗ (A). Then (X, Σ, µ) is a measure space. Proof. It is trivial to verify that ∅ is in Σ and that Σ is closed under complementation. In order to verify countable additivity we ﬁrst verify ﬁnite additivity. Suppose that A, B ∈ Σ.

Finally we deﬁne is ∞ Eab . Ea = b=a By the basic convexity property we have b P r(Eab ) ≤ P r({s : | m=a S m2 1 − | > }). 2 m 2 Now using Chebyshev’s inequality just as we did in the proof of the weak law we get P r(Eab ) b 1 . 2 m m=a 1 ≤ 2 4 As we let b go to inﬁnity and use the fact that for ﬁxed a the family Eab is a nested increasing family of sets, so that by continuity we get 1 P r(Ea ) = lim P r(Eab ) ≤ 2 b− →∞ 4 ∞ 1 . m2 m=a What does it mean for a sequence to be in a set of the form Ea ?

### Denumerable Markov Chains by John G. Kemeny, J. Laurie Snell, Anthony W. Knapp

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