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.

Mar 21, 2024 · However, the sigma-algebra in general is a larger than the topology, so is there any chance that a map that brings measurable sets to measurables sets is a measurable map ?

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...

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, …

From this viewpoint, the construction of the measure (as well as the σ σ -algebra of measurable sets) is just a generalization of the natural construction of the Riemann integral on Rn R n - you try to …

Jan 4, 2019 · I know for sure that if f2 f 2 is measurable, that doesn't imply that f f is measurable, but how does the condition:

The first observation to make here is there's a difference between a measurable space (that is, a set paired with a σ σ -algebra) and a measure space (that is, a measurable space equipped with a …

Dec 24, 2022 · I suspect that the author actually meant that for a nonnegative measurable f f the expression ∫ fdμ ∫ f d μ is always well defined (eventually taking value +∞ + ∞). This cannot be said of …