Thanks for the clarification about directionality of limits. This is like an analysis flashback. In all the continuous cases I ever deal with, everything’s the same either way. But when you get into step functions, the limits are different depending on which way you approach. I’ll have to be more careful about the definitions, which will take a while, because I’ll have to understand them better.

Good point about different measures on the same algebra. It’d make sense that you’d think about the conditionals as another measure, because they’ll satisfy all the properties.

Again, I’m really just used to thinking about everything in terms of joint densities of fairly simple functional forms.

]]>Also, measurability depends only on the sigma-algebra, not the measure (which turns out to be important in probability theory, where, with conditional probabilities, you have many different measures on the same sigma-algebra).

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