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I have some difficulty in determining the measurability in product space. Suppose the product space is [itex]T \times \Omega [/itex] equipped with [itex] \mathcal{T} \otimes \mathcal{F}[/itex] where [itex] ( T , \mathcal{T} , \mu ), ( \Omega , \mathcal{F} , P)[/itex] are themselves measurable spaces.

Now, if there exists a set [itex] T_0 [/itex] in [itex] T [/itex] with [itex] \mu(T_{0}^{c}) =0[/itex] and, for each fixed [itex] t \in T_0 [/itex], a property holds almost everywhere in [itex] \Omega [/itex], so this means there exists a [itex] \Omega_{t} [/itex] such that [itex] P(\Omega_{t}^{c}) = 0 [/itex] and that property holds on this set.

How can we conclude that the property will holds almost everywhere in the product space [itex]T \times \Omega [/itex]? Are they saying the set [itex]T_0 \times\Omega_{t} [/itex] is measurable?

Or in other words, when does the measurability hold if the second set [itex]\Omega_{t} [/itex] is a function of the first set [itex]T_0 [/itex]?

Thanks very much.

Wayne

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# Measurability in Product Space

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