Prove this function is $\mathcal{B}(\mathbb{R}^d)$ measurable

Fix an $r>0$. \begin{align*} x\rightarrow\dfrac{\mu(B_{r}(x))}{m(B_{r}(x))} \end{align*} is a lower semi-continuous function. Then, $f$ as the supremum of a class of lower semi-continuous functions, is still a lower semi-continuous function, and hence a Borel function.

Control the sectional curvature

Can $ I_n(z) = (\frac{z}{2})^n\sum_{k=0}^\infty\frac{(-1)^k}{k!(n+k)!}\frac{1}{k+(n+k)+1}(\frac{z}{2})^{2k}$ be expressed by the Bessel function?

Finding the norm of the linear functional $f(x)=\int_{-1}^0 x(t) \, dt - \int_0^1 x(t) \, dt$ on $C[-1,1]$

How many solutions does $f'(x)=1/2$ have, given that $|f(x)-f(y)|\ge|x-y|$?

If a student who received an A in probability is chosen at random, what is the probability that he/she also received an A in calculus?

Calculating Radical of an Ideal [closed]

Is it possible to express $\frac{1}{n}+\frac{1}{1+n}+\dots+\frac{1}{2n-1}$ as a sum

Showing there is a node in the graph with only one edge

How to simply prove that a certain graph is a subdivision of $K_{3,3}$

proof by definition of limit of $\lim_{x\to -\infty}\tanh(x)=-1$

Determining $A$ such that $\lim \limits_{x \to\infty }(\sqrt{Ax^2+2x}−5x)$ exists and is finite

$f\left(x_{0}\right)=\sup _{x \in X} f(x)<\infty$ in a compact metric space