Integrating over a power of the infinitesimal

Question

I don't know if the title makes sense (or if the question makes sense at all for that matter) but here I go.

Suppose I have a piecewise constant function $y=f(x)$ with $x,y\in\mathbb{R}^+$, described as follows \begin{align} y = \left\{ \begin{array}{lr} y_1 : x \in [0,x_1)\\ y_2 : x \in [x_1,x_2)\\ \ \vdots\\ y_n : x \in [x_{n-1},x_n) \end{array} \right.\end{align}

Now for such a function I define a quantity called $C$ that is given by $$ C = \sum_{i=1}^n y_i(x_i - x_{i-1})^k $$ where $k \in (1,2)$.

I want to be able to compute this quantity for arbitrary positive-valued functions that have finitely many discontinuities on the interval $[0,x_n)$. I cannot figure out how to go from the summation above to integration.

I have the feeling that if I define a measure for a closed interval $[a,b]$ as $(b-a)^k$ that might lead to something but then how do I compute the integral for a simple function like $y = x$? I am a bit lost. I would really appreciate some help.

Answer 1

As Peter Franek observes in the comments, your formulation will not lead to a useful object, because it will turn out that any limiting process on partition will give you a result of zero.

The reason for the above, as well as the proper formulation of your intuition, is the mechanism of Hausdorff measure. The $k$-Hausdorff measure is an outer measure on $\mathbb{R}^n$ and other metric spaces which is usually supported on sets of Lebesgue measure zero (unless $k=n$, when it gives Lebesgue measure for measurable sets) and is an important tool for distinguishing the 'size' of measure zero sets (which, otherwise, are just... zero measure sets!).

For a gorgeous introduction to the Hausdorff measure and the associated dimension concept, read Gerald Edgar's Measure, Topology and Fractal Geometry. It is accessible, well written and contains a lot of nice concepts, incuding a proper framework for the intuition you are trying to formalize in your post.