Journey into the realm of measure theory and you will confront profound questions: Can every subset of the plane possess a surface area? Can every conceivable function be integrated? What truly defines a random variable? These innocent queries open the Pandora’s box of measure theory, a cornerstone in mathematics with far-reaching implications. Measure theory serves as the bedrock for two expansive domains: Partial Differential Equations and Probability Theory. Without its principles, many Partial Differential Equations would remain unformulated, let alone proven to be well-posed. Moreover, measure theory provides the rigorous framework for modern probability theory, offering insights into real-valued random variables, conditional expectation, and stochastic processes. In essence, understanding measure theory is akin to unlocking the foundational structures that underpin these critical areas of mathematics.