Definition of Integral in Multiple Variables

[Astrum]

Dyadic Cube $$C_{k,N} = X \in\ \mathbb{R}^{n} \frac{k_{i}}{2^{N}} \leq x_{i} < \frac{k_{i}+1}{2^{N}} for 1 \leq i \leq n$$

Where $$k = \pmatrix { k_{1} \cr k_{2} \cr \vdots \cr k_{i} \cr }$$

I understand that N is the level of the cubes, but what does k equal?

I'm having trouble visualizing this in my head.

$A \subset \mathbb{R}^{n}$

$$M_{A}(f)= supp_{x \in A}f(x); m_{A}(f) = inf_{x \in A}f(x)$$ $$U_{N}(f) = \sum M_{c}(f) vol_{n}C$$ $$L_{N}(f) = \sum m_{c}(f) vol_{n}C$$

I get the general idea, but I can't really see this in my head.

If you take the supp value of a function on a given cube, and multiply it by the volume of you cube, you get volume again??

I can't really get the image straight.

I get that this is just an extension of single variable, so it means that U must equal L which must equal I (integral).

 

[Mandelbroth]

Dyadic Cube $$C_{k,N} = X \in\ \mathbb{R}^{n} \frac{k_{i}}{2^{N}} \leq x_{i} < \frac{k_{i}+1}{2^{N}} for 1 \leq i \leq n$$

Where $$k = \pmatrix { k_{1} \cr k_{2} \cr \vdots \cr k_{i} \cr }$$

I understand that N is the level of the cubes, but what does k equal?

[...]

If you take the supp value of a function on a given cube, and multiply it by the volume of you cube, you get volume again??

Let's see if we can work this out.

What does k do in your dyadic cube relation? What does it act on?

As for your second question, what is the supremum of a function in this case?

I think the answers are in asking those questions.

 

[Astrum]

Well, from what I understand, the supf(x) is the maximum value of the function on a given cube, which in single variable gives a height to the rectangle. In multiple dimensions, I can't "see" it. I assume that this also gives.

As for what k means, I guess it means the number of cubes under the graph?

I can't find any other sources explaining this, perhaps my book just has a weird way of defining the multiple integral.