Measure Theory is a branch of mathematics that deals with the concept of size or volume in a rigorous way. It extends the idea of measuring lengths, areas, and volumes to more complex sets, allowing mathematicians to analyze functions and spaces that are not easily quantifiable. This theory is essential for understanding concepts in probability, integration, and functional analysis. At its core, Measure Theory introduces the notion of a measure, which assigns a non-negative value to subsets of a given space, capturing their "size." This framework helps in defining integrals and probabilities in a consistent manner, making it a foundational tool in advanced mathematics and its applications in fields like statistics and physics.