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By Lynn H. Loomis

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Let A = {p :j(p) =pt. 0}. If A is a null set, then ncp A is a null function and ncpA j aocp A, so that the function equal to +ao on A and zero elsewhere (oocpA) is a null function. But 0 ~ If I ~ oocpA so that f is null. Conversely, iff is null, then so is (ni f I) n 1, and, since (ni f I) n 1 i cpA, it follows that cpA is a null function and A is a null set. We are now in a position to remark on an ambiguity remaining in 12F, namely, that, since the values ±ao are allowed for a summable function, the sum / 1 2 of two summable functions is not defined on the set where/1 = ±ao and/2 = =Fao.

I+ J I](/) I~ J

E: L,/and where + oo is allowed as a possible value of I. It will be shown in 12C that this number is independent of the particular sequence I/.. } converging tof. Assuming this, it is evident that the new definition agrees with the old in case/ E: L (for then we can setf,. = /) and that the extended I satisfies (1) and (2) with c ~ 0. It wilt also follow from 12C that I satisfies (3'). 12C. Lemma. ). Proof. We first notice that, if k E: L and lim/,. ) ~ I(k), for f,. ~ f,. ) ~ lim I(f,. n k) = I(k), by (3') and (4).

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An Introduction to Abstract Harmonic Analysis by Lynn H. Loomis


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