A Borel measure is a type of measure defined on the Borel σ-algebra, which consists of all open sets in a given topological space. It provides a systematic way to assign a size or volume to sets, allowing for the integration of functions over these sets. The most common example of a Borel measure is the Lebesgue measure, which extends the concept of length, area, and volume to more complex sets. Borel measures are essential in real analysis and probability theory, as they help in defining and working with measurable functions and sets. They satisfy properties like countable additivity, meaning the measure of a union of disjoint sets equals the sum of their measures. This makes Borel measures a foundational concept in modern mathematics.