chapter  16
A difference between the Riemann and Lebesgue iterated integrals
ByGilbert W. Bassett Jr., Roger Koenker
Pages 9

If one is given a real-valued function f which is defined on a nondegenerate rectangle [ a , b ] × [ c , d ] ⊂ R 2 $[a,b] \times [c,d] \subset \mathbf{R}^2$ , then one may consider the question of the existence of two iterated integrals ∫ c d ( ∫ a b f ( x , y ) d x ) d y , ∫ a b ( ∫ c d f ( x , y ) d y ) d x $$\int _c^d(\int _a^bf(x,y)dx)dy,~~~~~\int _a^b(\int _c^df(x,y)dy)dx$$