A bounded function $f:Q\to\mathbb R$ defined on the rectangle $Q$ in $\mathbb R^n$ is integrable if the lower integral of $f$ is equal to the upper integral of $f$. The integral of $f$ over $Q$ is defined to be \(\int_Q f = \underline{\int_Q} f = \overline{\int_Q} f.\)
Alternatively, let $f: S\to \mathbb R$ be a bounded function from the bounded set $S$ to $\mathbb R$, and let $f_S$ be the extension of $S$ in the usual sense to $\mathbb R$. Choose $Q$ a rectangle containing $S$. The integral of this function over $S$ is \(\int_S f = \int_Q f_S,\) if it exists.
Wikidata ID: Q697181