$Set theory question$

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$Set theory question$

I am reading J. Neveu's "Mathematical foundations of the calculus of probability." There he presents what he calls a distributivity formula: $$\bigcup_{j \in J} \bigcap_{i \in I_j} F_i^j=\bigcap_{\{i_j\} \in K}\bigcup_{j \in J} F_{i_j}^j$$ where $K=\Pi_{j\in J}I_j$ (the set of all sequences \{$i_j, j \in J$\}). My question is regarding the notation. The LHS is quite clear to me. However, I'm now sure what are actually the members of K. Say $J=\{1,2,\ldots\}$ and $I_j=\{1,2,\ldots\}$. What is actually $i_j$ here? I'd appreciate an illumination on the RHS here and what is actually going on there..

Set Theory

Dannyboy990

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In your example, if $I_j = \mathbb{N}$ for each $j \in J$, and $J = \mathbb{N}$ as well, then $K = \mathbb{N}^\mathbb{N}$.

In general, $K$ is the product of the $I_j$, so essentially for each element of $J$ you're choosing an element in the corresponding $I_j$. An element of $K$ is a $J$-tuple of elements such that the $j$-th element belongs to $I_j$; alternatively, it is a function $f \colon J \rightarrow \coprod_{j \in J} I_j$ (the disjoint union) such that $f(j) \in I_j$.

The RHS is doing as such: take an element of $K$, say a function $f$ in the notation above. Then you take the union of the sets $F_{f(j)}^j$, and then you take the intersection for each function of those sets. What you're doing is finding a way to associate each $j$ to some $i_j \in I_j$, so that you can pick a subcollection $F_{f(j)}^j$ of the sets $F_i^j$ that is indexed by $j$ only.

If you can give me a choice of the sets $F_i^j$, I can write down an example, if this is not clear.

Alessandro Iraci

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Dannyboy990

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Thanks. I think I am beginning to understand. For simplicity's sake, let's assume j={1,2} and I_j={1,2,...}. What are the members in $I_1 \times I_2$?
Alessandro Iraci

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That's just NxN: you have to pick one element from I_1 and one element from I_2, so you get pairs of positive integers.
Dannyboy990

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I accidentally hit the accept instead of submitting another comment :) So bear with me here. So ${i_j} \in K$ is a combination of a member from $I_1$ and from $I_2$. So what is confusing to me is what is the $j$ indexing doing there?
Alessandro Iraci

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Well, the j indexing is the index of the family I_j. So, if J={1,2}, you have I_1 x I_2, but if J = {1,2,3} you'd get I_1 x I_2 x I_3 instead. If your element was, say, {i_j} = (4,7,1) \in NxNxN, you'd have i_1 = 4, i_2 = 7, i_3 = 1.
Dannyboy990

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Ok. So that was confusing to me as {i_j} could as well be (1,1,1) in the example you gave, and the $j$ index is just to say that it is a $J$-tuple? But if ${i_j}$ is a $j$-tuple, what is F_{i_j}^j? How can you index F by a $j$-tuple? Hope that makes sense.
Dannyboy990

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Ok I think I understand. So ${i_j}$ denotes the sequence and $i_j$ denotes a member from $I_j$, right?
Alessandro Iraci

0

Yes that's correct! And F_{i_j}^j is just some set, it doesn't really matter.
Dannyboy990

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Thanks :)
Alessandro Iraci

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You're welcome!

$Set theory question$

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