Pi system

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In mathematics, a π-system on a set Ω is a set P, consisting of certain subsets of Ω, such that

  • P is non-empty.
  • A ∩ B ∈ P whenever A and B are in P.

Contents

[edit] Properties

[edit] Uniqueness of Extension

A finite measure may be considered to be uniquely determined by its values on a π-system in the following sense. Let μ and υ be measures on (X, Σ) and \mathcal{I} be a π-system that generates Σ. If

  • \mu ( X ) = \nu ( X ) < \infty
  • \mu |_{\mathcal{I}} = \nu |_{\mathcal{I}}

then μ=υ.

[edit] See also

[edit] References

  • Allan Gut, 2005. Probability: A Graduate Course. Springer.
  • David Williams, 2007. Probability with Martingales. Cambridge Mathematical Textbooks.
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