Alphabeta MathHow statement and proof provenance work
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Convergence in measure
Definition
Let be a measure space and let be measurable. The sequence converges to in measure when for every real ,
Thus convergence in measure asks only that, for each fixed threshold , the measure of the bad set vanish as .
Depends on
Used by
- The spikes k chi_(0,1/k) converge almost everywhere and in measure to zero but not in L¹ Counterexample
- The translates of the unit interval converge almost everywhere to zero but not in measure Counterexample
- Cauchy sequences in measure Definition
- The typewriter sequence converges in measure and in L¹ but nowhere pointwise Example
- FALSE: almost-everywhere convergence implies convergence in measure on every measure space False statement
- FALSE: convergence in measure implies almost-everywhere convergence False statement
- On a finite measure space, the truncated L¹ metric metrises convergence in measure Proposition
- Almost uniform convergence implies almost-everywhere convergence and convergence in measure Theorem
- Cauchy sequences in measure converge in measure Theorem
- Convergence in L¹(mu) implies convergence in measure Theorem
- Convergence in measure determines the limit almost everywhere Theorem
- On a finite measure space, almost-everywhere convergence implies convergence in measure Theorem
- Riesz's subsequence theorem for convergence in measure Theorem
- Vitali convergence theorem on finite and sigma-finite measure spaces Theorem
Dependency tree · two levels
2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- H. L. Royden and P. M. Fitzpatrick, Real Analysis, 4th ed., Section 5.2 (standard reference, not scraped)
- Terence Tao, 245A Notes 4: Modes of convergence (standard reference, not scraped)