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.

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 …

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...

What's your definition of measurability and w.r.t. what $\sigma$-algebras? Because from the point of view of the theory of general measurable spaces, your problem is the definition of measurability. …

Mar 12, 2016 · Using this, one can easily show that a Baire measurable homomorphism from a Baire group to a separable group is continuous (Pettis' theorem). See Kechris, Classical Descriptive Set …

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.

Oct 29, 2019 · And any other constructions of sets based in complementation and countable union of measurable sets defines also an element of $\mathcal {A}$, that is, a measurable set.

Feb 23, 2017 · A measurable function (might need to be bounded or of bounded variation - not sure!) is approximately continuous i.e. continuous except on a set of measure 0. Measurability is quite a …

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 $\mu$.