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Let $X = \bigcup_{j=1}^{n}A_j$, where $A_j \neq \emptyset$. Let $\mathcal{F}$ be a $\sigma$-algebra generated by $A_1,\dots, A_n$. Show that $F \in \mathcal{F}$ if and only if $$ F = \bigcup_{(\epsilon_1,\dots,\epsilon_n)\in J} A_1^{\epsilon_1} \cap \dots \cap A_n^{\epsilon_n} $$ where J is some set of zero-one sequences of length n, and $A_j^0 = A_j$ and $A_j^1 = A_j^C$. Estimate from below and above cardinality of $\mathcal{F}$.

My attempt (only $\implies$):

Let $F \in \mathcal{F}$ and $x \in F$ and $M_x$ to be an interesection of all sets $A_1, \dots, A_m$ such that $x \in A_n$(what I mean is $M_x = \bigcap A\in \bigcup_{j=1}^nA_j, A_j \ni x$). Then for arbitrary: $$ \{x\} \subset \left( A_1 \cap \dots \cap A_m \right) = M_x $$ but if we denote those sets A which contain x with indexes $1, \dots, m$ (this is not strictly formal as we don't really rearrange sets here) then: $$ \{x\} \subset \left( A_1 \cap \dots \cap A_m \cap A_{m+1}^C \cap \dots \cap A_n^C \right) \subset \left( A_1 \cap \dots \cap A_m \right) = M_x $$ because $x \notin A_{m+1}, \dots, A_n$.

We may now take $\bigcup_{x\in F}$, and create a zero-one sequences in the process:

  • if $x \in A_j$ then $\epsilon_j = 0$
  • if $x \notin A_j$ then $\epsilon_j = 1$ (we take a complement)

$$ F = \bigcup_{x\in F}x \subset \bigcup_{(\epsilon_1,\dots,\epsilon_n)\in J} A_1^{\epsilon_1} \cap \dots \cap A_n^{\epsilon_n} $$

But that is just a half of proof $\implies$. I need to show: $$ F \supset \bigcup_{(\epsilon_1,\dots,\epsilon_n)\in J} A_1^{\epsilon_1} \cap \dots \cap A_n^{\epsilon_n} $$ As well as $\impliedby$ and also estimate cardinalities. Could I get some hints? Thanks.

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1 Answer 1

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Hints:

1) Sets of the form $A^{\epsilon_1} \cap A^{\epsilon_1}\cap...\cap A^{\epsilon_n}$ form a partition of $X$ (which means they are pair-wise disjoint and their union is $X$).

2) Each $A_i$ is a union of sets from this partition; for example $A_1$ is simply the union of all sets of the form $A^{\epsilon_1} \cap A^{\epsilon_1}\cap...\cap A^{\epsilon_n}$ where $\epsilon_1=1$.

3) Using 2) show that sigma algebra generated by $A_i$'s is same as those generated by the sets of this partition.

4) For any finite partition $\{B_1,B_2,...,B_m\}$ the sigma algebra generated is nothing but all possible unions of the sets $B_i$.

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