Let f:Q→Rf:Q\to\mathbb Rf:Q→R be a bounded function defined on the rectangle QQQ in Rn\mathbb R^nRn. The upper integral of fff over QQQ is the infimum over all partitions of QQQ of the upper sum of QQQ: ∫Q‾f=infPU(f,P)\overline{\int_Q}f = \inf_P U(f,P)∫Qf=PinfU(f,P)