In the error list of FitzPatrick’s Advanced Calculus, it’s said that on p. 479, ln. 14–21, “it doesn’t seem that $\vect{P}_k$ can be chosen as indicated…” I spent some time to understand what $\vect{P}^*_k$ was.
It took me an hour to see what’s wrong with $\vect{P}_k$. It’s much easier to understand the words using figures.
What the $\vect{J}$’s are?
Suppose $\vect{I} \equiv [0,2] \times [0,2]$.
$\vect{J}$ is any of the small rectangles in the above figure inside $\vect{I}$.
Understanding $\vect{P}_k (\vect{J})$
$\vect{P}_k (\vect{J})$ is a partition of $\vect{J}$. In other words, $\vect{P}_k (\vect{J})$ contains a small black solid square and the gray dashed lines inside the square.
The above figure can’t be a partition of $\vect{I}$.
Understanding $\vect{P}^*_k (\vect{J})$
$\vect{P}^*_k (\vect{J})$ is a refinement of $\vect{J}$.