The Riemann integral is of course well-defined for continuous functions $f$ over a closed interval $[a,b]$. We can trivially extend it to finite disjoint unions of such intervals. I ask if there's a sensible way to extend it to any compact subset of the real line.
Here's my idea. The connected components of a compact subset of $\mathbb R$ are closed intervals, potentially trivial. At most countably of these will be nontrivial, as each of these contains a unique rational point. If we enumerate these intervals as $[a_n,b_n]$, by boundedness, the length $b_n-a_n$ must tend to zero, meaning that the limit $$\sum_{n=0}^\infty\int_{a_n}^{b_n}f$$ exists and is independent of the enumeration of intervals.
My questions are:
- Is this construction standard? If so, is there a reference?
- Is this a well-behaved notion? As in, do the standard properties like linearity, monotonicity, etc. hold, or does something go awry?
- Is there a generalization of this idea to higher dimensions? Can one integrate continuous functions over any compact subset of $\mathbb R^n$, and does Fubini's theorem hold?
There already is a generalization of the Riemann integral beyond closed, bounded intervals to arbitrary bounded sets in $\mathbb{R}$ (or beyond closed, bounded hyperrectangles in $\mathbb{R}^n)$. Given a bounded set $S \subset \mathbb{R}$ which need not be closed and an interval $[a,b] \supset S$, the Riemann integral over $S$ is defined as
$$\int_S f(x) \, dx = \int_a^bf_S(x) \, dx,$$
where $f_S(x) = \begin{cases} f(x), & x \in S \\ 0, & \text{otherwise} \end{cases}$
and this definition can be shown to be independent of the choice of enclosing interval $[a,b]$.
The generalized integral -- if it exists -- has the same properties as the conventional Riemann integral. For example, it can be defined in terms of upper and lower Darboux integrals.
Furthermore, the Lebesgue criterion of integrability holds, stating that the Riemann integral exists if and only if the set of discontinuities of $f_S$ has measure zero. This is where it gets trickier than when the integral is defined over closed intervals. Even if $f$ is continuous almost everywhere in $S$, the integral over $S$ can fail to exist if $S$ is not Jordan-measurable, that is the boundary of $S$ has measure $0$.
If the boundary of $S$ fails to have measure zero, then unless $f(x) \to 0$ from inside $S$ on every subset of $S$ with non-zero measure, the Riemann integral of $f_S$ over $[a,b]$ will not exist. In that case, by definition, the integral of $f$ over $S$ will not exist.
Computing the integral over an arbitrary subset of $\mathbb{R}$ is no longer amenable to such devices as finding an antiderivative. We have to rely on finding or more likely estimating a limit of Riemann sums or an infimum of upper Darboux sums over all partitions of $[a,b]$.