A solution to Pfeffer’s problem. (English) Zbl 1021.26003

On the square \(S=[0,1]\times [0,1]\) a function \(f\) is given for which \[ \int_a^b \left(\int_c^d f(x,y) dy\right) dx = \int_c^d\left(\int_a^b f(x,y) dx\right)dy \] for each subinterval \([a,b]\times [c,d] \subset S\) while the integral \(\int_S f\) does not exist. The above equality can be taken assuming that the integrals are Lebesgue integrals. The function \(f\) constructed in the paper is even not Kurzweil integrable on the square \(S\).


26B15 Integration of real functions of several variables: length, area, volume
26A39 Denjoy and Perron integrals, other special integrals