The construction of novel measures in measure theory is often done using Carathéodory’s Extension Theorem, though this can be a very tedious process. In a 1918 paper by Percy Daniell, he introduced...Show moreThe construction of novel measures in measure theory is often done using Carathéodory’s Extension Theorem, though this can be a very tedious process. In a 1918 paper by Percy Daniell, he introduced what are now known as Daniell integrals on vector lattices. The extension of this integral to a larger space naturally leads to a measure-constructing process. Given topological spaces X with some kind of projective structure, we introduce a strategy using projective systems for developing a Daniell integral on a vector lattice related to X, which in turn yields a measure on X. We apply this alternative strategy of constructing measures to some examples, where in particular we construct measures on infinite product spaces as well as infinite dimensional separable Hilbert spaces.Show less