Hereditary system generated by

Stub grade: A*

This page is a stub

This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:

Demote grade once content is put in place, I really want to pull the "smallest" note out and give it a name (generated system of sets?) as this occurs a lot, and can be applied any time the type of system in question is closed under arbitrary intersection, that is the intersection of an arbitrary family of a type of system is a system of that type in and of itself.

Warning:This page is little more than notes at the moment, however everything stated here is verified and correct

Definition

The hereditary system generated by a collection of sets, $S$ , which we denote: $\mathcal{H}(S)$ is the smallest^{[Note 1]} hereditary system containing $S$ ^[1].

Claim 1: $\mathcal{H}(S)=\{V\in\mathcal{P}(T)\ \vert\ T\in S\}$ ^{[Note 2]} where $\mathcal{P}(A)$ denotes the power set of $A$ .

Proof of claims

Grade: A

This page requires one or more proofs to be filled in, it is on a to-do list for being expanded with them.

Please note that this does not mean the content is unreliable. Unless there are any caveats mentioned below the statement comes from a reliable source. As always, Warnings and limitations will be clearly shown and possibly highlighted if very important (see template:Caution et al).
The message provided is:

Pretty routine proof by contradiction. I've done it on some paper somewhere

Notes

Jump up ↑
We do not mean smallest in the sense of cardinality arguments, we also do not mean smallest in the sense of ⊂ relation (as given any two hereditary systems containing S we cannot be sure that either one is a subset (proper or not) of the other! Instead we mean smallest in the following sense:
- $\mathcal{H}(S):=\bigcap_{\text{All hereditary systems of sets, }\mathcal{H}\text{, where }S\subseteq\mathcal{H} }\mathcal{H}$
This is extremely similar to sigma-ring generated by and many other generators involving systems of sets - there is certainly something to abstract here.
Grade: A
This page requires one or more proofs to be filled in, it is on a to-do list for being expanded with them.
Please note that this does not mean the content is unreliable. Unless there are any caveats mentioned below the statement comes from a reliable source. As always, Warnings and limitations will be clearly shown and possibly highlighted if very important (see template:Caution et al).
The message provided is:
Prove that the intersection of an arbitrary family of hereditary systems of sets (among other systems of course!) is itself a hereditary system of sets
Jump up ↑
There are many ways to write this and this may not be the best. The "defining property" (Note to self: explore notion between FOL sentences and sets) is:
- $\left[A\in\mathcal{H}(S)\right]\iff[\exists B\in S(A\in\mathcal{P}(B))]$

References

Jump up ↑ Measure Theory - Paul R. Halmos

[1] Jump up ↑ We do not mean smallest in the sense of cardinality arguments, we also do not mean smallest in the sense of $\subset$ relation (as given any two hereditary systems containing $S$ we cannot be sure that either one is a subset (proper or not) of the other! Instead we mean smallest in the following sense:
$\mathcal{H}(S):=\bigcap_{\text{All hereditary systems of sets, }\mathcal{H}\text{, where }S\subseteq\mathcal{H} }\mathcal{H}$
This is extremely similar to sigma-ring generated by and many other generators involving systems of sets - there is certainly something to abstract here.
Grade: A
This page requires one or more proofs to be filled in, it is on a to-do list for being expanded with them.
Please note that this does not mean the content is unreliable. Unless there are any caveats mentioned below the statement comes from a reliable source. As always, Warnings and limitations will be clearly shown and possibly highlighted if very important (see template:Caution et al).
The message provided is:
Prove that the intersection of an arbitrary family of hereditary systems of sets (among other systems of course!) is itself a hereditary system of sets

[2] $\mathcal{H}(S):=\bigcap_{\text{All hereditary systems of sets, }\mathcal{H}\text{, where }S\subseteq\mathcal{H} }\mathcal{H}$

[3] Jump up ↑ There are many ways to write this and this may not be the best. The "defining property" (Note to self: explore notion between FOL sentences and sets) is:
$\left[A\in\mathcal{H}(S)\right]\iff[\exists B\in S(A\in\mathcal{P}(B))]$

[4] $\left[A\in\mathcal{H}(S)\right]\iff[\exists B\in S(A\in\mathcal{P}(B))]$

[MTH-2] Jump up ↑ Measure Theory - Paul R. Halmos

[Note 1]

[1]

[Note 2]

Hereditary system generated by

Contents

Definition

Proof of claims

Notes

References

Navigation menu

Views

Personal tools

Navigation

Search

Tools