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Then In ~ L:~ hi and L:~ /(hi) < 1(/n) + E. Setting h = L:~ hi, we have h E: U and /(h) = L:~ /(hi) by 12D. Moreover,/~ hand /(h) ~ lim 1(/n) + E. ,. ~ f ~ h and /(h) - /(g) < 2E. ). 12H. A family of real-valued functions is said to be monotone if it is closed under the operations of taking monotone increasing and monotone decreasing limits. The smallest monotone family including L will be designated ffi and its members will be called Baire functions. If h ~ k, then any monotone family ~n which contains (g U h) n k for every g E: L also contains (/ u h) n k for every I E: ffi, for the functions/ such that (/ U h) n k E: mr form a monotone family which includes L and therefore includes ffi.

EQUIVALENCE AND MEASURABILITY 13A. If A c S, then "IPA" conventionally designates the characteristic function of A: 'PA(P) = 1 if p E: A and 'PA(P) = 0 if p E: A'. (A) as /('PA). } is a disjoint sequence of integrable sets, then Ui A,. (An). (A) < oo, then A will be said to be summable. The integral I will be said to be bounded if S is summable: it is clear that I is then a bounded linear functional with respect to the uniform norm. 13B. We shall now further restrict L by adding a hypothesis used by Stone [47], namely, that JE:L =>Jn 1 E:L.

J. is weakly closed. 1. , that is, that there exists an Xo E: MJ. such that Po(xo) ~ 0. , there exists E and xh · · ·, x .. such that no G E: M satisfies I G(x,)- P 0 (x,) I< E for every i = 1, · · ·, n. The elements x, map M onto a subspace of complex Euclidean n-space G ~ (G(x 1), • • · , G(x .. ) ), and no point of this subspace is in an E-cube about the point (P0 (x 1) • • ·, Po(x.. )). In particular the subspace does not contain this point and is of dimension at most n - 1, so that there exist constants ch · · ·, c" such that E c,G(x,) = 0 for every G E: M and E c£0 (x,) = 1.

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