Let P\mathbb{P}P be the set of all possible partitions of the interval [a,b][a, b][a,b]. Upper Integral U(f)=inf{ U(f;P);P∈P }=∫abf‾U(f)=\inf{\set{U(f;P);P\in\mathbb{P}}}=\overline{\int_{a}^{b}{f}} U(f)=inf{U(f;P);P∈P}=∫abf Lower Integral L(f)=sup{ L(f;P);P∈P }=∫abf‾L(f)=\sup{\set{L(f;P);P\in\mathbb{P}}}=\underline{\int_{a}^{b}{f}} L(f)=sup{L(f;P);P∈P}=∫abf For a bounded function fff, always L(f)≤U(f)L(f)\le U(f)L(f)≤U(f)