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## Homework Statement

Let [tex]\mathscr{X}[/tex] be a set, [tex]\mathscr{F}[/tex] a [tex]\sigma-[/tex]field of subsets of S, and [tex]\mu[/tex] a probability measure on [tex]\mathscr{F}[/tex]. Suppose that [tex]A_{1},\ldots,A_{n} [/tex] are independent sets belonging to [tex]\mathscr{F}[/tex]. Let [tex]\mathscr{F}_{k}[/tex] be the smallest subfield of [tex]\mathscr{F}[/tex] containing [tex]A_{1}, \ldots, A_{k}[/tex]. Show that if [tex]A \in \mathscr{F}_{k}[/tex], then [tex]A, A_{k+1}, \ldots, A_{n}[/tex] are indepdendent.

## Homework Equations

Two sets are independent iff [tex]\mu(A \cap B) = \mu(A)\mu(B)[/tex].

## The Attempt at a Solution

Really, my question here is what the smallest field is. It seems that, given a set [tex]\mathscr{X}[/tex], the smallest field [tex]\mathscr{F}_{s}[/tex] containing it is simply [tex]\left\{\emptyset, \{ \mathscr{X}\}\right\}[/tex]. Am I just crazy?

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