Matthew wrote:

> As a note, Anindya remarks in comment #22 of Lecture 62, there may be issues with that lattice identity I have tried to use...

I don't see any problem with that. Suppose we have a poset \\(\mathcal{V}\\) with all joins, and a doubly indexed family of elements in \\(\mathcal{V}\\), say \\( \lbrace v_{a b} \rbrace_{a \in A, b \in B} \\) where \\(A\\) and \\(B\\) are arbitrary sets. Then

\[ \bigvee_{a \in A} \bigvee_{b \in B} v_{ab} = \bigvee_{a \in A, b \in B} v_{ab} = \bigvee_{b \in B} \bigvee_{a \in A} v_{ab} \]

It takes a bit of work to show this, but it seems so obvious I don't have the energy right now. All three are just different ways of thinking about the least upper bound of all the elements \\(v_{a b}\\).

I guess in real analysis one should have seen this sort of fact when \\(\mathcal{V} = [-\infty,\infty] \\) is the [extended real number line](https://en.wikipedia.org/wiki/Extended_real_number_line).

This fact implies a related fact for sums of arbitrary collections of numbers in \\([0,\infty]\\):

\[ \sum_{a \in A} \sum_{b \in B} v_{ab} = \sum_{a \in A, b \in B} v_{ab} = \sum_{b \in B} \sum_{a \in A} v_{ab} \]

The reason is that these sums are defined as suprema of partial sums.

> As a note, Anindya remarks in comment #22 of Lecture 62, there may be issues with that lattice identity I have tried to use...

I don't see any problem with that. Suppose we have a poset \\(\mathcal{V}\\) with all joins, and a doubly indexed family of elements in \\(\mathcal{V}\\), say \\( \lbrace v_{a b} \rbrace_{a \in A, b \in B} \\) where \\(A\\) and \\(B\\) are arbitrary sets. Then

\[ \bigvee_{a \in A} \bigvee_{b \in B} v_{ab} = \bigvee_{a \in A, b \in B} v_{ab} = \bigvee_{b \in B} \bigvee_{a \in A} v_{ab} \]

It takes a bit of work to show this, but it seems so obvious I don't have the energy right now. All three are just different ways of thinking about the least upper bound of all the elements \\(v_{a b}\\).

I guess in real analysis one should have seen this sort of fact when \\(\mathcal{V} = [-\infty,\infty] \\) is the [extended real number line](https://en.wikipedia.org/wiki/Extended_real_number_line).

This fact implies a related fact for sums of arbitrary collections of numbers in \\([0,\infty]\\):

\[ \sum_{a \in A} \sum_{b \in B} v_{ab} = \sum_{a \in A, b \in B} v_{ab} = \sum_{b \in B} \sum_{a \in A} v_{ab} \]

The reason is that these sums are defined as suprema of partial sums.