Upper & Lower Integral

Let $\mathbb{P}$ be the set of all possible partitions of the interval $[a, b]$ .

Upper Integral

U(f)=\inf{\set{U(f;P);P\in\mathbb{P}}}=\overline{\int_{a}^{b}{f}}

L(f)=\sup{\set{L(f;P);P\in\mathbb{P}}}=\underline{\int_{a}^{b}{f}}

For a bounded function $f$ , always $L(f)\le U(f)$

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November 2, 2026 1 min read

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