This book provides a solid foundational knowledge of measure theory by presenting the key components in detailed, easily understandable steps and connecting them with numerous examples. Many exercises of varying difficulty levels support the understanding of the material. Solutions to all exercises are provided in the appendix for self-assessment. Applications of measure theory in stochastic processes are demonstrated in chapters on conditional expectations and likelihood functions. The required prior knowledge is kept to a minimum, as necessary fundamentals from set theory and the theory of real numbers are reviewed and deepened at the beginning in a clear format.