ABSTRACT
Interesting results arise when the measure and the topology are ‘compatible’ in some sense. Measure theory on topological spaces developed its own constructions using such compatibility conditions and these constructions differ to some extent from those of abstract measure theory, as for example in the classical Daniell construction. This chapter provides some necessary preliminaries concerning the extension of mappings defined on lattices of sets and investigates the measures on Hausdorff spaces. It explains the continuous extended real-valued function and formulates Lusin’s theorem which describes the relationship between continuity and measure. The chapter discusses the topological properties of Hausdorff spaces and the congruence invariance of the n-dimensional Lebesgue measure.
