# Complete measure

From Encyclopedia of Mathematics

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

#### Comments

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.

#### References

[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: http://www.encyclopediaofmath.org/index.php?title=Complete_measure&oldid=16648

This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098