Geometric measure theory uses measure properties to study the geometric properties of sets in various spaces (usually Euclidean space).
Here are some key notions:
1) The Coarea formula expresses the integral of a function over an open set in Euclidean space, and is an adaptation of Fubini's theorem to geometric measure theory
2) Radon Measures are a type of measure on the $\sigma -$algebra of Borel sets of a Hausdorff topological space that is finite locally and inner regular. Radon measures are sets with the least 'regularity' required to approximate tangent spaces
3) There is the concept of the varifold, which is a measure-theoretic form of a differentiable manifold. It does this by maintaining the algebraic structure, but replacing any differentiability requirements with ones provided by rectifiable sets (these are sets that are smooth in the measure-theoretic sense).