Let (FP) abbreviate the statement that ${\int }_0^1\left({\int }_0^{1}fdy\right)dx={\int }_0^1\left({\int }_0^{1}fdx\right)dy$ holds for every bounded function f: [0,1]² → ℝ whenever each of the integrals involved exists. We shall denote by (SFP) the statement that the equality above holds for every bounded function f: [0,1]² → ℝ having measurable vertical and horizontal sections. It follows from well-known results that both of (FP) and (SFP) are independent of the axioms of ZFC. We investigate the logical connections of these statements with several other strong Fubini type properties...