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Sample text

Chapters 9 and 11, on this possibility).  This is made precise by Lebesgue, using the already defined outer measure, as follows.  A set is called Lebesgue measurable iff μ*(E)=μ*(E).  First, by the subadditivity of μ*, we have satisfies the functional equation (13).  But holds as stated.  Lebesgue showed, using his μ*, μ* , that the class of Lebesgue measurable sets forms a σ­algebra containing the Borel class properly.

The argument in both cases is similar, involving a nontrivial idea which becomes an important tool in the subject.  Thus let is a π­class, consider as in (5), (7) Since by construction Page 26 Now to see that in (7), consider the new class (8) It follows from the preceding that ­class, and (b) follows.  Let More generally, the σ­algebra generated by all the open sets of a topological space is termed its Borel σ­algebra.  Observe that (9) and hence that the σ­algebras generated by these classes are identical, because Similarly, (10) and so On the other hand, one has and from which one deduces at once that (11) Hence This establis hes all assertions.

Hence μ(Ak )=μ(Bk )+μ(Ak−1), so that we have Thus μ(Bk )=μ(Ak )−μ(Ak−1), μ(Ak−1)<∞ , (2) and by σ­additivity (ii): Now so that the sequence again has a limit, since (3) Given and (4) Cancelling the finite number on both sides of (4), one has (ii).  For instance, let μ(∙)=m(∙), the Lebesgue measure on Page 23 So 0=μ(limn An )

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