View source for Properties of classes of sets closed under set-subtraction

{{Refactor notice|grade=A|msg=The old page was crap, it is preserved below but it doesn't provide anything useful}}
==Statement==
Suppose {{M|\mathcal{A} }} is an arbitrary class of [[set|sets]] with the property that:
* {{M|1=\forall A,B\in\mathcal{A}[A-B\in\mathcal{A}]}} where {{M|A-B}} denotes "[[set subtraction]]" ({{AKA}}: [[relative complement]])
Then we claim the following are true of {{M|\mathcal{A} }}:
# {{M|\mathcal{A} }} is {{M|\cap}}-closed. That is: {{M|1=\forall A,B\in\mathcal{A}[A\cap B\in\mathcal{A}]}}
==Proof of claims==
{{Requires proof|easy=1|grade=C|msg=Outlines follow for each claim:
# Prove {{M|1=A\cap B=A-(A-B)}} or indeed {{M|B-(B-A)}}. Include picture of [[Venn diagram]]
}}
==Notes for further claims==
I know from a reference (see the old page below) that:
# If {{M|\mathcal{A} }} is {{sigma|{{M|\cup}}-closed}} then it is {{sigma|{{M|\cap}}-closed}}
# Any countable union of sets in {{M|\mathcal{A} }} can be expressed as a countable disjoint union of sets in {{M|\mathcal{A} }}<ref group="Note">Note that this doesn't require {{M|\mathcal{A} }} to be closed under union, we can still talk about unions we just cannot know that the result of a union is in {{M|\mathcal{A} }}. This simply says IF there is a [[sequence]] of elements of {{M|\mathcal{A} }}, say {{MSeq|A_n|in=\mathcal{A} }}, where {{M|1=\bigcup_{n=1}^\infty A_n}} just happens to be in {{M|\mathcal{A} }} then there exists a sequence of [[pairwise disjoint]] sets whose union is the same.</ref>
These feel very "measure theory specific" though and might be true in the case of being closed under arbitrary union or for an arbitrary collection of sets whose union is {{M|\mathcal{A} }}
==Notes==
<references group="Note"/>
==References==
<references/>
{{Elementary set theory navbox|plain}}
{{Measure theory navbox|plain}}
{{Theorem Of|Measure Theory|Elementary Set Theory}}








=OLD PAGE=
==Theorem statement==
If {{M|\mathcal{A} }} is a class of subsets of {{M|\Omega}} such that<ref name="PTACC">Probability Theory - A comprehensive course - Second Edition - Achim Klenke</ref>
* <math>\forall A,B\in\mathcal{A}[A-B\in\mathcal{A}]</math> - that is ''closed under [[Set subtraction|set-subtraction]]'' (or {{M|\backslash}}-closed)
Then we have:
# {{M|\mathcal{A} }} is {{M|\cap}}-closed
# {{M|\mathcal{A} }} is {{Sigma|{{M|\cup}}-closed}}{{M|\implies}} {{M|\mathcal{A} }} is {{Sigma|{{M|\cap}}-closed}}
# Any countable union of sets in {{M|\mathcal{A} }} can be expressed as a countable disjoint union of sets in {{M|\mathcal{A} }}<ref group="Note">Note that this doesn't require {{M|\mathcal{A} }} to be closed under union, we can still talk about unions we just cannot know that the result of a union is in {{M|\mathcal{A} }}</ref>
{{Begin Theorem}}
Proof:
{{Begin Proof}}
{{Todo|Page 3 in<ref name="PTACC"/>}}
{{End Proof}}{{End Theorem}}
==Notes==
<references group="Note"/>

==References==
<references/>

{{Theorem Of|Measure Theory}}