/sci/ - Science & Math

[Anonymous No.8619733]

Why is the product σ-algebra of [math]\mathbb R^T[/math] (as, e.g., given by Kolmogorov's extension theorem) not "rich"?

sauce for the "not rich" statement: lecture + https://en.wikipedia.org/wiki/Kolmogorov_extension_theorem + https://fabricebaudoin.wordpress.com/2012/03/24/lecture-2-measure-theory-in-function-spaces/

[Anonymous No.8619752]

Have you tried taking the functions in the Excercises (of what I think is the only time "rich" appears in that article) and use the operations of a sigma-algebra to see if you find a contradiction (something of sigma algebras doesn't work for all those functions).

Why R^T and not R^[0,1] or whatever is talked about there

You'll not get much of an answer here.

Also I hate Wikipedia articles on stochastic related topics, there the fact that each page has it's own notation/consistency/perspective really hurts.

[Anonymous No.8619800]

>>8619752
My guess (never had a proper measure theory course because fuck my university's curriculum) is that "[math]\prod_{x\in [0,1]}(-\infty,1] [/math]" cannot be obtained by countably many intersections (/unions) of cylinder sets where only countably many "coordinates" are not [math](-\infty,\infty)[/math], but Borel sets instead. So we can only measure events that can be written knowing only countably many values, e.g. subsets of continuous/cadlag function spaces. Is that what is meant there?

T to keep it general. In our lecture we'll go on to deal with pretty arbitrary (though real-valued I think) Gaussian processes.

At this point pretty much any answer is enough to make my day so thank you anon. Pretty new to this stuff, do you have any alternatives to Wikipedia for quickly looking up definitions/theorems I'm unfamiliar with?

Picture for the rest of /sci/.