This Wikipedia article says the following about Fubini's theorem:
If a function is Lebesgue integrable on a rectangle $X\times Y$, then one can evaluate the double integral as an iterated integral:$$\iint \limits _{X\times Y}f(x,y)\,{\text{d}}(x,y)=\int _{X}\left(\int _{Y}f(x,y)\,{\text{d}}y\right){\text{d}}x=\int _{Y}\left(\int _{X}f(x,y)\,{\text{d}}x\right){\text{d}}y.$$Here all integrals are Lebesgue integrals. Fubini's theorem is not true as stated for the Riemann integral, but it is true if the function is assumed to be continuous on the rectangle, and sometimes this weaker result is called Fubini's theorem in multivariable calculus.
It states, without a counterexample, that Fubini's theorem is not true as stated for the Riemann integral. Does anyone know an example where $f(x,y)$ is Riemann-integrable but Fubini's theorem fails?