A Borel measure is a way to assign a size or volume to sets in a mathematical space, particularly in real analysis. It is defined on the Borel σ-algebra, which consists of all open sets and can be generated by them through countable unions, intersections, and complements. This measure helps in understanding the properties of sets, such as their size and how they relate to each other. 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 probability theory, where they help define probabilities for events in a continuous sample space.