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Bookmark or Share- measure theory - Infinite product of measurable spaces - Mathematics ...Because of this, I believe that if you make a category out of all measurable spaces and measurable functions in the obvious way, the product $\sigma$-algebra you have defined turns out to be the categorical product: i.e., it satisfies the requisite universal mapping property. Again, this is the situation for the product topology.
- measure theory - Definition of a measurable function? - Mathematics ...So at the end of the day, to check that a real-valued function is measurable, by definition we must check that the preimage of a Borel measurable set is measurable. But this boils down, as shown above, to proving that $\{x \mid f(x) > \alpha \} = f^{-1}( (\alpha, \infty)) \in \Sigma$ for all $\alpha \in \mathbb{R}$, since this implies that the ...
- A function that is measurable but not Lebesgue integrable.The function $1/x$ on $\mathbb{R}$ (defined arbitrarily at $0$) is measurable but it is not Lebesgue integrable. In general, a function is Lebesgue integrable if and only if both the positive part and the negative part of the function has finite Lebesgue integral, which is not true for $1/x$.
- measure theory - What does it mean by $\mathcal{F}$-measurable ...I always see this word $\mathcal{F}$-measurable, but really don't understand the meaning. I am not able to visualize the meaning of it. Need some guidance on this. Don't really understand $\sigma(Y)$-measurable as well. What is the difference?
- How to determine if a random variable is $\\mathcal F$-measurable?$\begingroup$ The definition of a random variable is a measurable real-valued function defined on a sample space. So to be more precise, you should ask something like "is this function measurable?" or "is this map a random variable?" $\endgroup$ –
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