A Borel measure is a way to assign a size or volume to certain sets in a mathematical space, particularly in real analysis. It is defined on the Borel σ-algebra, which consists of all sets that can be formed from open intervals through countable unions, intersections, and complements. This measure helps in understanding properties of functions and sets in a rigorous way. 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.