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Hereditary sigma-ring generated by

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Contents

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1 Definition
2 Claims
3 See also
4 Notes
5 References

Definition

Given a system of sets, $S$ , the hereditary $\sigma$ -ring generated by $S$ is^[1]:

The smallest hereditary $\sigma$ -ring containing $S$

We denote this $\mathcal{H}_{\sigma R}(S)$ ^{[Note 1]}

Claims

$\mathcal{H}(\sigma_R(S))=\sigma_R(\mathcal{H}(S))$ - see Notes:Hereditary sigma-ring

See also

Hereditary system generated by

Notes

Jump up ↑ If $\mathcal{H}(A)$ denotes the hereditary system of sets generated by $A$ and $\sigma_R(B)$ the sigma-ring generated by $B$ , then we have $\mathcal{H}(\sigma_R(A))=\sigma_R(\mathcal{H}(A))$

References

Jump up ↑ Measure Theory - Paul R. Halmos

v • d • e Measure Theory

Overview of the basic and important objects studied in measure theory

Set-structures	ring of sets, algebra of sets, $\sigma$ -ring, $\sigma$ -algebra, hereditary $\sigma$ -ring measurable space

Measures	Pre-measure $\leadsto$ outer-measure $\leadsto$ measure (see Extending pre-measures to measures (via: Extending pre-measures to outer-measures $\rightarrow$ "Restricting outer-measures to measures" maybe?). Alternatively: inner-measure can be used. measure space

Functions	measurable map, simple function (standard representation)

Spaces

Integrals	Integrating a simple function $\leadsto$ Integral of a positive function $\leadsto$ Integrating all measurable functions

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