Problem. Let $f: A \to \mathbb{R}$ be a function defined in $m$-dimensional block $A$, such that $m \leq f(x) \leq M$ for any $x \in A$. Given a partition $P$ of $A$, there is a partition $Q$ of the $(m+1)$-dimensional block $A \times [m,M]$ such that $S(f;P) - s(f;P)$ is the sum of volumes of the blocks of $Q$ containing the graph of $f$.
This is the first question of my problem list. The point is that I don't know what to use to start solving. So far I don't have many results to use. I know that if $B$ is an $(m+1)$-dimensional block so, $$\operatorname{Vol}B = \prod_{1}^{m+1}(b_{i}-a_{i})$$ where $[a_{i},b_{i}]$ are the edges of the block. It seems to me that I must define the partition $Q$ using the partition $P$, but I have no ideia. Can someone help me?
For a partition $P$ of $A$ with subrectangles in $\{R_j\}_{j=1}^n$ , we have
$$S(f,P) - s(f,P) = \sum_{j=1}^n (M_j - m_j) vol(R_j),$$
where $m_j = \inf_{x \in R_j} f(x)$ and $M_j = \sup_{x \in R_j} f(x)$.
For each subrectangle $R_j$ construct a higher dimensional rectangle $R_j \times [m_j,M_j]$. Note that
$$vol(R_j \times [m_j,M_j]) = (M_j - m_j)vol(R_j)$$
Finally create a partition $Q$ of $A \times [m,M]$ as $P \times P'$ where $P'$ is a partition of $[m,M]$ formed from all of the points $m_j, \,M_j$ ($1 \leqslant j \leqslant n$).
The original rectangles $R_j \times [m_j,M_j]$ will be decomposed as a union of rectangles from the partition Q, and I'll leave it to you to show that this meets your requirements.