Maharam's theorem

Maharam's theorem

In mathematics, Maharam's theorem is a deep result about the decomposability of measure spaces, which plays an important role in the theory of Banach spaces. In brief, it states that every complete measure space is decomposable into "non-atomic parts" (copies of the unit interval [0,1] on the reals), and "purely atomic parts", using the counting measure on some discrete space.[1] The theorem is due to Dorothy Maharam.

The result is important to classical Banach space theory, in that, when considering the Banach space given as an Lp space of measurable functions over a general measurable space, it is sufficient to understand it in terms of its decomposition into non-atomic and atomic parts.

References

  1. ^ D. Maharam, "On homogeneous measure algebras", Proceedings of the National Academy of Sciences USA, 28 1942, pp. 108-111.

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