A first return examination of the Lebesgue integral. (Q595810)

It is proven that if \(f:I^n \to {\mathcal R}\) is a Lebesgue-integrable function then there is a countable dense set \(D\) and an enumeration \((x_p)\) of \(D\) such that for each \(\eta > 0\) there is a \(\delta > 0\) such that if \({\mathcal P}\) is a partition of \(I^n\) having norm less than \(\delta \), then \[ \sum_{J\in {\mathcal P}}f(r(J))| J| - \int_{I^n}f| < \eta , \] where \(r(J)\) denotes the first element of \((x_p)\) that belongs to \(J\). This shows that a Lebesgue integrable function comes equipped with a sequence of points which one can use in conjunction with a simple ``first return-Riemann'' integration procedure to compute the integral.