Apr 17, 2016 · As I have always understood it (and various online references seem to go with this tradition) is that when one says a function is increasing or strictly increasing, they mean it is …

Could you guide me how to prove that any monotone function from R → R R → R is Borel measurable? Since monotone functions are continuous away from countably many points, …

Let f be a monotone function on the open interval (a,b). Then f is continuous except possibly at a countable number of points in (a,b). Assume f is increasing. Furthermore, assume (a,b) is bou...

Note that the Monotone Convergence Theorem applies regardless of whether the above interpretations: a non-decreasing (or strictly increasing) sequence converges if it is bounded …

A function is convex if and only if its gradient is monotone. Ask Question Asked 9 years, 9 months ago Modified 1 year, 7 months ago

Jun 2, 2020 · I can understand that if $f$ is monotone, then $g$ is monotone by continuous inverse theorem. But is this really necessary for the inverse function theorem to be used?

May 24, 2020 · I am watching a very nice set of videos on measure theory, which are great. But I am not clear on what the motivation is behind the monotone convergence theorem--meaning …

In words, it is a monotone class containing the algebra $\mathcal A$. Since $\mathcal M$ is the smallest monotone class containing $\mathcal A$, it must be contained in any other monotone …

Show that a divergent monotone increasing sequence converges to +∞ + ∞ in this sense. I am having trouble understanding how to incorporate in my proof the fact that the sequence is …

Mar 6, 2015 · This is claimed to be named as "Beppo Levi's theorem". It is not stated that fn f n are non-negative or non-negative a.e., so it is clearly not the same assumptions of Monotone …