15. (a) A finite family B₁, ie I of o-algebras is independent iff for every choice of non-negative Bi-measurable random variable Y₁, i € I, we have E(ΠΥ;) = ΠΕ(;). - iel iel (One direction is immediate. For the opposite direction, prove the result first for positive simple functions and then extend.) (b) If {B,,t e T) is an arbitrary independent family of o-algebras in (S2, B, P), the family (B, te T) is again independent if B, B₁, (t e T). Deduce from this that {f(X), t e T) is a family of independent random variables if the family {X,, te T) is independent and the f, are measurable. In order for the family (X₁, t e T) of random variables to be independent, it is necessary and sufficient that

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