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We introduce smooth L^\infty differential forms on a singular (semialgebraic)set X in R^n. Roughly speaking, a smooth L^\infty differential form is acertain class of equivalence of 'stratified forms', that is, a collection ofsmooth forms on disjoint smooth subsets (stratification) of X with matchingtangential components on the adjacent strata and bounded size (in the metricinduced from R^n). We identify the singular homology of X as the homology ofthe chain complex generated by semialgebraic singular simplices, i.e.continuous semialgebraic maps from the standard simplices into X. Singularcohomology of X is defined as the homology of the Hom dual to the chain complexof the singular chains. Finally, we prove a De Rham type theorem establishing anatural isomorphism between the singular cohomology and the cohomology ofsmooth L^\infty forms.

De Rham Theorem for L^\infty forms and homology on singular spaces