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Riemann's criterion on a nondegenerate rectangle in $R^{m}$ : integrability is equivalent to arbitrarily small Darboux gaps

Statement

A bounded $f : Q \to R$ on a nondegenerate rectangle is Riemann integrable if and only if, for every $ε > 0$ , some grid $P$ satisfies $U (f, P) - L (f, P) < ε$ .

Facts & Assumptions

Given: A bounded function on a nondegenerate rectangle.

[L1]

The lower and upper integrals are the supremum and infimum in The lower and upper Darboux integrals over a nondegenerate rectangle in $R^{m}$ .

[L2]

Near-supremum and near-infimum elements exist (Epsilon characterisation of the supremum, Epsilon characterisation of the infimum).

[L3]

Common refinements improve both sums (Refinement raises multidimensional lower sums and lowers upper sums, with a quantitative boundary-slab estimate, Grid partitions of a rectangle in $R^{m}$ , their cells, refinements and mesh).

Proof

technique · direct

1.1L1L2L3

If the two integrals equal $I$ , choose $P_{-}$ with $L (f, P_{-}) > I - ε / 2$ and $P_{+}$ with $U (f, P_{+}) < I + ε / 2$ . A common refinement $P$ has gap below $ε$ .

1.2L1L3given

Conversely, a common refinement shows every lower sum is at most every upper sum, so for every $P$ , $0 \leq \overline{\int_{Q}} f - \underline{\int_{Q}} f \leq U (f, P) - L (f, P)$ . Arbitrarily small gaps force the integral difference to be $0$ .

Riemann's criterion on a nondegenerate rectangle in $R^{m}$ : integrability is equivalent to arbitrarily small Darboux gaps

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

Riemann's criterion on a nondegenerate rectangle in RmRm: integrability is equivalent to arbitrarily small Darboux gaps

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

Riemann's criterion on a nondegenerate rectangle in $R^{m}$ : integrability is equivalent to arbitrarily small Darboux gaps