In particular, what does $$H_{ij}=I_iJ_j$$ mean? It says that $H_{ij}$ are a partition of $[a,b]$. But it also says that both $I_i$ and $J_j$ are partitions of $[a,b]$.
So, what does the author mean by this notation, and hy are $H_{ij}$ a partition of $[a,b]$. Is it the intersection of the two intervals $I_i$ and $J_j$?. Like a product of indicator variables maybe?
EDIT:
Please someone answer. Im guessing its a simple thing for a mathematician, but I have never seen this notation.
It likely means $H_{ij} = I_i \cap J_j$, so that $\{H_{ij}\}$ becomes a partition that is a refinement of both $\{I_i\}$ and $\{J_j\}$. This is similar to the notation $AB$ for the intersection of events $A \cap B$ in probability.
Side note: the author's use of "$\epsilon$" for both an actual epsilon as well as the inclusion "$\in$" is giving me conniptions.