On Weak Compactness in Spaces of Measures

@article{Zhang1997OnWC,
  title={On Weak Compactness in Spaces of Measures},
  author={Xiaodong Zhang},
  journal={Journal of Functional Analysis},
  year={1997},
  volume={143},
  pages={1-9}
}
Abstract It is proved that a weak* compact subsetAof scalar measures on aσ-algebra is weakly compact if and only if there exists a nonnegative scalar measureλsuch that each measure inAisλ-continuous (such a measureλis called a control measure forA). This result is then used to obtain a very general form of the Vitali–Hahn–Saks Theorem on finitely additive vector measures. Finally, it is proved that a weak* compact subsetAof regular Borel measures on anF-space is weakly compact if and only if… 
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References

SHOWING 1-10 OF 10 REFERENCES

Banach Lattices and Positive Operators

I. Positive Matrices.- 1. Linear Operators on ?n.- 2. Positive Matrices.- 3. Mean Ergodicity.- 4. Stochastic Matrices.- 5. Doubly Stochastic Matrices.- 6. Irreducible Positive Matrices.- 7. Primitive…

Measures on F-spaces

A variant of Grothendieck's Theorem on weak* convergent sequences

A note on order bounded vector measures

Factorization of vector measures

Banach spaces of continuous functions

``Linear Operators,'' Part I, Interscience

  • New York London,
  • 1958

``Topological Vector Spaces,'' 5th ed., Springer-Verlag

  • New York Berlin Heidelberg Tokyo,
  • 1986

``Sequences and Series in Banach Spaces,'' Springer-Verlag

  • New York Berlin,
  • 1984

``Vector Measures,'' Mathematical Surveys, Number 15

  • Am. Math. Soc., Providence,
  • 1977