A measure is a convenient mathematical object (function) that can represent the positions of strawberries in a field, distribution of water in the ocean, or probabilities of winning over the lottery numbers — the measure counts the number of strawberries in a given area, reports the amount of water in a certain sea, and evaluates the probability of a lottery ticket to win. This abstract unifying framework enables one to rigorously ‘measure’ quantities over a space, and also enable integration. It also allows elegant notation for probability theory. Here we briefly describe key ideas of measure theory without proof. This material is mostly based on references  and .

To define a measure, we need a measurable space (X, F); a non-empty set X and a σ-algebra F on X. Here X is the space where our stuff to be measured lies, and F gives special structure of the space such that things are well defined and pathological sets can be avoided. An algebra F of  X is a set of subsets of X such that it contains the empty set, and closed under set union and complement. A σ-algebra is an algebra that is closed under countable union. Elements in F are said to be measurable.

A measure μ on (X, F) is a non-negative extended real valued function on F that is countably additive; $\mu \left( \cup_i^\infty A_i \right) = \sum_i^\infty \mu (A_i)$

where $A_i$ are disjoint measurable sets. Additivity makes sense because we want $\mu(A) + \mu(B) = \mu(A \cup B)$ when A and B are disjoint (number of strawberries should add up for different fields). If μ is always finite valued, then it is called a finite measure. For a special case, a probability measure is a finite measure where μ(X) = 1.

Given a set E, we denote the smallest σ-algebra that contains E as σ(E) and say σ(E) is the σ-algebra generated by E. A measure μ on F is determined by its values on any algebra that generates F (Carathéodory extension theorem).

A predicate P(x) holds almost everywhere μ (or μ-a.e. for short) if it is true except for a set of measure zero, that is, $\mu(E) = 0, \forall x \in E^{c}, P(x)$.

A function f from a measurable space (X, F) to a measurable space (Y, G) is measurable if $\forall E \in \mathcal{G}, f^{-1}(E) \in \mathcal{F}$For a topological space (X, U), the σ-algebra generated by the open sets is called the Borel σ-algebra (or Borel set). Borel set links the measurable functions and continuous functions — every continuous function from X to the real line is measurable with respect to the Borel algebra. When the topological space is induced by a metric, Borel set and Baire set coincide. Baire set is the smallest σ-algebra with respect to which the continuous functions are measurable.

Real-valued Borel-measurable functions are closed under algebraic operations and limit. Moreover, they can be approximated by limit of simple functions. A simple function is a finite linear combination of indicator function of measurable sets. By the linearity of integration, integration of a Borel-measurable function with respect to a measure μ can be defined by letting the integration of an indicator function on the measurable set A as μ(A): $\int \mathbb{I}_{A} \mathrm{d}\mu = \mu(A)$.

References:

1. Halmos, P. R. Measure theory. Springer-Verlag, 1974
2. Daley, D. J. & Vere-Jones, D. An Introduction to the Theory of Point Processes. Springer, 1988

P.S. Terry Tao has a great summary on measure theory and integration.