During my class in real analysis, my teacher mentioned an alternative method to prove Riesz's theorem for $C([0,1])$. He just mentioned the method, but did not prove it. (A usual way of doing this is provided here: Riesz representation theorem for $C([0,1])$) Step 1: Let $\Lambda$ be a nonnegati...

I discovered that the following integrals are equal: $$ \int_0^1sx^{s-1}\exp\bigg(\frac{t}{\log(x)}\bigg)~dx=\int_0^1\exp\bigg(\frac{st}{\log(x)}\bigg)~dx $$ Let $f^s(x)=x^s,$ then the LHS can be written as $$ \int_0^1\frac{d}{dx}\bigg(x^s\bigg)\exp\bigg(\frac{t}{\log(x)}\bigg)~dx$$ It reminds me...

I remember a video, perhaps a Numberphile episode where a mathematician was described who would simply move in to the home of a collaborator with which they were engaged to so that they could work more closely together, essentially sleeping on their sofa and going to work with them and keeping a ...

$S = \mathbb{Z}, a * b = a+b^2$ Commutative: $a*b = b*a$ $a*b = a + b^2$ and $b*a = b+a^2$ and they aren't the same at all. Associative: $(a*b)*c = a*(b*c)$ $(a*b)*c = (a+b^2)* c = a+b^2+c^2$ and $a*(b*c) = a + (b+c^2)^2$ and they aren't the same at all. therefore it's not a binary operation on t...

Is the following claim true? Let $(X, \mathcal{B}, \mu)$ be a measure space with $\mu$ nonatomic. If $0 < k < \mu(X)$, there exists an increasing sequence $B_1, B_2, \dots \in \mathcal{B}$ (i.e., $B_1 \subset B_2 \subset \cdots $) with $\lim_{n \to \infty}\mu(B_n) = k$. I'm honestly not sure if...