A Solution to Pfeffer’s Problem

Abstract

We give an example of a non-integrable function $f$ on $[0,1]\times [0,1]$ such that $$ \int_{\alpha}^{\beta} (\int_{\gamma}^{\delta} f(x,y)\,dy)\,dx =\int_{\gamma}^{\delta} (\int_{\alpha}^{\beta} f(x,y)\,dx)\,dy$$ for each subinterval $[\alpha,\beta] \times [\gamma, \delta]$ of $[0,1] \times [0,1]$.