A general integral
Volume 483 / 2012
Abstract
We define an integral, the distributional integral of functions of one real variable, that is more general than the Lebesgue and the Denjoy–Perron–Henstock–Kurzweil integrals, and which allows the integration of functions with distributional values everywhere or nearly everywhere.
Our integral has the property that if $f$ is locally distributionally integrable over the real line and $\psi\in\mathcal{D}( \mathbb{R}) $ is a test function, then $f\psi$ is distributionally integrable, and the formula \[ \langle \mathsf{f},\psi\rangle =(\mathfrak{dist}) \int_{-\infty}^{\infty}f(x) \psi(x) \,\mathrm{d}x, \] defines a distribution $\mathsf{f}\in\mathcal{D}^{\prime}( \mathbb{R}) $ that has distributional point values almost everywhere and actually $\mathsf{f}(x) =f(x) $ almost everywhere.
The indefinite distributional integral $F(x) =( \mathfrak{dist}) \int_{a}^{x}f( t) \,\mathrm{d}t$ corresponds to a distribution with point values everywhere and whose distributional derivative has point values almost everywhere equal to $f( x) .$
The distributional integral is more general than the standard integrals, but it still has many of the useful properties of those standard ones, including integration by parts formulas, substitution formulas, even for infinite intervals (in the Cesàro sense), mean value theorems, and convergence theorems. The distributional integral satisfies a version of Hake's theorem. Unlike general distributions, locally distributionally integrable functions can be restricted to closed sets and can be multiplied by power functions with real positive exponents.
