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An integrable function on the unit square with one Dirichlet section and only one defined order of ordinary iteration
Example
On , define Then is Riemann integrable with integral . Every horizontal section is integrable, but the vertical section at is the Dirichlet function. Hence the -then- ordinary iterated integral exists and equals , whereas the other ordinary order is not defined.
Facts & Assumptions
Given: The displayed function on the unit square.
The Dirichlet function is the indicator of the rationals, with values and on two dense sets (The Dirichlet function , and Thomae's function with at a rational in lowest terms with and at every irrational ).
The published boundedness false statement computes the Dirichlet function's unequal lower and upper Darboux integrals, so boundedness alone does not give integrability (FALSE: every bounded function on is Riemann integrable).
A bounded function is Riemann integrable when grids make its upper-minus-lower sum arbitrarily small (Riemann's criterion on a nondegenerate rectangle in : integrability is equivalent to arbitrarily small Darboux gaps).
Riemann--Fubini distinguishes ordinary iterated integrals from lower/upper completion across exceptional sections (Riemann--Fubini on product rectangles, with lower and upper section integrals and content-zero exceptional sections).
Verification
Every lower sum of is . Put grid lines immediately to either side of with total intervening width below ; only cells in that strip can have supremum , so the upper sum is below . Thus [L3] gives integrability and integral .
For fixed , the -section is either zero or the indicator of the singleton , and in both cases its integral is . The resulting outer function is zero, so that ordinary iteration exists and equals .
For fixed , the -section is zero; at it is the Dirichlet function and is nonintegrable by [L2]. Therefore the reverse ordinary iteration is undefined, exactly as [L4] permits despite multiple integrability.
Depends on
- Riemann--Fubini on product rectangles, with lower and upper section integrals and content-zero exceptional sections
- The Dirichlet function $1_{\mathbb{Q}}$, and Thomae's function $t$ with $t(x) = 1/q$ at a rational $x = p/q$ in lowest terms with $q \ge 1$ and $t(x) = 0$ at every irrational $x$
- FALSE: every bounded function on $[a,b]$ is Riemann integrable
- Riemann's criterion on a nondegenerate rectangle in $\mathbb{R}^m$: integrability is equivalent to arbitrarily small Darboux gaps
Used by
Nothing in the library uses this result yet.
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Sources
- J. Lebl, Basic Analysis II, Example 10.2.1 (standard reference, not scraped)