By H. Royden, P. Fitzpatrick

ISBN-10: 7111313054

ISBN-13: 9787111313052

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Thus (9) It remains to prove this inequality in the opposite directon. According to Proposition 6, for each natural number n, m(U Ek) = ±m(Ek). k=l k=l Since U~l Ek contains Uk=l Ek, by the monotonicity of outer measure and the preceding equality, m(U Ek) > ±m(Ek) for each n. k=l k=l The left-hand side of this inequality is independent of n. : ~ m(Ed. (10) D From the inequalities (9) and (10) it follows that these are equalities. According to Proposition 1, the outer measure of an interval is its length while according to Proposition 2, outer measure is translation invariant.

It is only natural to ask if, in fact, there are any sets that fail to be measurable. The answer is not at all obvious. We know that if a set E has outer measure zero, then it is measurable, and since any subset of E also has outer measure zero, every subset of E is measurable. This is the best that can be said regarding the inheritance of measurability through the relation of set inclusion: we now show that if E is any set of real numbers with positive outer measure, then there are subsets of E that fail to be measurable.

The proof of the following proposition follows directly from the definition of u-algebra. Proposition 13 Let:F be a collection of subsets of a set X. Then the intersection A of all u-algebras of subsets of X that contain :F is a u-algebra that contains :F. Moreover, it is the smallest u-algebra ofsubsets of X that contains :F in the sense that any u-algebra that contains :F also contains A. Let {An}~l be a countable collection of sets that belong to au-algebra A. Since A is closed with respect to the formation of countable intersections and unions, the following two sets belong to A : The set limsup{An}~l is the set of points that belong to An for countably infinitely many indices n while the set lim inf{ An} ~l is the set of points that belong to An except for at most finitely many indices n.

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Real Analysis by H. Royden, P. Fitzpatrick

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