There are multiple reinforcement learning (RL) benchmarks (i.e. a set of environments where we can test our RL algorithms), for example, the DeepMind Control Suite. However, given that I am currently not particularly familiar with them or their details, I don't really know if any of them comes wi...

Let $(X,S,\mu)$ be a measure space. Suppose $f\in \mathcal{L}^p$. Let $\epsilon>0$. Show that there exists a set $E \in S$ with $\mu(E)<\infty$ such that if $F\in \mathcal{S}$ and $F\cap E=\varnothing$ then $||f\chi_{F}||_p<\epsilon$ My idea was to do something like: $f\in \mathcal{L}^p$ implies ...

I calculated that the number of rank $k$ matrices in $\mathbb{F}_q^{n \times m}$ (where we may assume $n \leq m$) for a prime power $q$ is equal to $$W_{n,m}(k) := \prod_{i = 1}^k \frac{(q^m - q^{i-1})(q^n - q^{i-1})}{(q^k - q^{i-1})}$$ And so summing all $W_{n,m}(k)$ from rank $k = 0$ to full ra...