Jul 3, 2020 · measurable functions provides a mathematics framework for what one would call "observables" in science (other than Mathematics, that is). The definition you presented, known as …

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.

There is no definition of "measurable set". There are definitions of a measurable subset of a set endowed with some structure. Depending on the structure we have, different definitions of …

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...

Suppose there is a family (can be infinite) of measurable spaces. What are the usual ways to define a sigma algebra on their Cartesian product? There is one way in the context of defining product

Jan 4, 2022 · Let's think about definitions. For a function to be measurable, the inverse image of open sets must be measurable. What is the domain of a measure? The domain is a sigma algebra. Thus, …

Nov 28, 2024 · Not every subset of a set of Borel measure $0$ is Borel measurable. Lebesgue measure is obtained by first enlarging the $\sigma$-algebra of Borel sets to include all subsets of set of Borel …

It is not true, in general, that the inverse image of a Lebesgue measurable (but not Borel) set under a continuous function must be Lebesgue measurable. The definition of a measurable function in …

Apr 29, 2021 · We need to show that for each $c\in \mathbb {R}$ this set is measurable. Now, if we consider $f|_ {A_c}:A_c\mapsto \mathbb {R}$, then we can use the fact that a bounded function …

On p. 6 of that textbook, it defines a $\mu$-measurable function as one which is measurable on the unique sigma algebra associated with the completion of the measure $\mu$.