Complete measure

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A measure on a -algebra for which and imply for every . Here is the total variation of ( for a positive measure).


Complete measures arise as follows (cf. [a1]). Let be a set, a -algebra of subsets of it and a positive measure on . It may happen that some set with has a subset not belonging to . It is natural, then, to define the measure on such a set as .

In general, let be the collection of all sets for which there exists sets such that , . In this situation, define . Then is a -algebra and becomes a complete measure on it (this process is called completion). is then called a complete measure space.


[a1] W. Rudin, "Real and complex analysis" , McGraw-Hill (1974) pp. 24
[a2] E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965)
How to Cite This Entry:
Complete measure. A.P. Terekhin (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098