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

May 18, 2017 · But not every measurable function is Borel measurable, for example no function that takes arguments from $ (\mathbb R,\ {\emptyset,\mathbb R\})$ is Borel measurable, because $\ …

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, …

May 5, 2021 · If $A\times B$ is a Lebesgue measurable set in $\mathbb R^ {n+m}$ and we have the additional condition that $\lambda_ {n+m} (A\times B)>0$, then we can conclude that $A$ and $B$ …

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 …

We are simply showing that the intersection of two measurable sets is again measurable. You are confusing properties of a measure function with what it means to be for a set to be measurable.