An Introduction to Measure Theory by Michael Fochler, Department of Mathematical Sciences, Binghamton University April 14, 2015   Measure theory is a branch of mathematics that provides a general framework for integration the same way that linear algebra provides a general framework for linear mappings between vector spaces and set topology provides a general framework for continuous functions. Examples of measures are length, area and volume, probabilities and integrals ∫ab f(x) dx when considered as functions of the domain of integration [a,b]. This talk will start with the most basic concepts of measure theory: σ-algebras, measurable functions, measures. Probabilistic interpretations will be given, such as the σ-algebra generated by a random variable X as the collection of all stochastically relevant information for X. Abstract integrals ∫ f d μ will be defined. The definition of the Lebesgue integral will be compared to that of the Riemann integral and the expected value of a random variable X will be defined as an abstract integral ∫ X d P. In the last part of the talk we shall examine conditioning with respect to sub-σ-algebras. We shall first deal with sub-σ-algebras generated by a countable measurable partition, examine the connection with conditional expectations and then give the general definition. We shall conclude the talk with the definition of martingales, the definition of the one-dimensional Wiener process and the (trivial) proof that the Wiener process is a martingale.