Abstract
Let (E,ε) be a measurable space. We shall say that a measure µ on (E,ε) is Σ-finite provided µ is a countable sum of finite measures. (This notation is due to Dynkin.) Obviously any σ-finite measure is Σ-finite. It is well known that the Fubini theorem is valid for Σ-finite measures, although most text books state it only for σ-finite measures. See, for example, Theorem 7.8a in [2] for the precise statement of what we shall mean by the Fubini theorem in this note. The statement of this theorem remains valid if σ-finite is replaced by Σ-finite. It is also well known that a translation invariant Σ -finite measure µ on R is a multiple of Lebesgue measure ρ; that is µ = cρ where 0 ≤ c ≤ ∞ For example, the proof of Theorem 60.A in [1] depends only on the Fubini theorem and so is valid for Σ-finite measures. (In [1] the much more general situation of Haar measure on a group is considered.)
Published Version
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