Convergence in Measure

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Let denote a measure space and let for . The sequence converges to in measure if for any . Furthermore, the sequence is Cauchy in measure if for every as [1]


Properties

  • If in measure and in measure, then in measure.[2]
  • If in measure and in measure, then in measure if this is a finite measure space. [2]

Relation to other types of Convergence

  • If in then in measure [1]
  • If in measure, then there exists a subsequence such that almost everywhere.[1]
  • If and measurable s.t. almost everywhere Then in measure.[3]

References

  1. 1.0 1.1 1.2 Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, second edition, §2.4
  2. 2.0 2.1 Craig, Katy. MATH 201A HW 8. UC Santa Barbara, Fall 2020.
  3. Craig, Katy. MATH 201A Lecture 18. UC Santa Barbara, Fall 2020.