Any help!Let F , G be σ-algebras on a set Ω. We define the join and meet of F and G to be F ∨ G := σ ( F ∪ G )and F ∧ G := F ∩ G ,respectively.However, how is F ∩ G defined? Is it (1) { A : A ∈ F and A ∈ G } or (2) { A ∩ B : A ∈ F and B ∈ G } ? In the theory of sets, I think it should be (1). But wouldn't it make much more sense to define F ∧ G to be (2)?Note that (2) is not a σ-algebra (or am I missing something?), but the σ-algebra generated by (2) is actually F ∨ G . On the other hand, (1) is itself a σ-algebra.

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Any help!Let       F    ,      G   be   σ-algebras on a set   Ω. We define the join and meet of       F   and       G   to be      F    ∨      G    :=  σ  (      F    ∪      G    )and      F    ∧      G    :=      F    ∩      G    ,respectively.However, how is       F    ∩      G   defined? Is it                    (1)                    {        A        :        A        ∈                  F                 and         A        ∈                  G                }            or                    (2)                    {        A        ∩        B        :        A        ∈                  F                 and         B        ∈                  G                }        ?            In the theory of sets, I think it should be (1). But wouldn&#039;t it make much more sense to define       F    ∧      G   to be (2)?Note that (2) is not a   σ-algebra (or am I missing something?), but the   σ-algebra generated by (2) is actually       F    ∨      G  . On the other hand, (1) is itself a   σ-algebra.

Athena Blanchard · Accepted Answer

Notation from lattice theory ...least upper bound:       F    ∨      G   is the smallest   σ-algebra       A   such that       F    ⊆      A   and       G    ⊆  A.greatest lower bound: F∧G is the largest   σ-algebra       A   such that       A    ⊆      F   and       A    ⊆  G.It turns out that the second one is merely the intersection, since the intersection of two   σ-algebras (on the same set) is a   σ-algebra. The first one is not (in general) the union, so it needs more study.We can do similar things for: subgroups of a given group; topologies on a given set; real-valued functions on a given set; and many more.

Any help! Let <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="

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