of importance: that of all Lebesgue measurable sets. This σ-algebra contains more sets than the Borel σ-algebra on and is preferred in integration theory, as it gives a complete measure space. Product σ-algebra Let and be two measurable spaces. The σ-algebra for the corresponding product space is called the product σ-algebra and is defined by Observe that is a π-system. The Borel σ-algebra for is generated by half-infinite rectangles and by finite rectangles. For example, For each of these two examples, the generating family is a π-system. σ-algebra generated by cylinder sets Suppose is a set of real-valued functions. Let denote the Borel subsets of A cylinder subset of is a finitely restricted set defined as Each is a π-system that generates a σ-algebra Then the family of subsets is an algebra that generates the cylinder σ-algebra for This σ-algebra is a subalgebra of the Borel σ-algebra determined by the product topology of restricted to An important special case is when is the set of natural numbers and is a set of real-valued sequences. In this case, it suffices to consider the cylinder sets for which is a non-decreasing sequence of σ-algebras. σ-algebra generated by random variable or vector Suppose is a probability space. If is measurable with respect to the Borel σ-algebra on then is called a random variable () or random vector (). The σ-algebra generated by is σ-algebra generated by a stochastic process Suppose is a probability space and is the set of real-valued functions on If is measurable with respect to the cylinder σ-algebra (see above) for then is called a stochastic process or random process. The σ-algebra generated by is the σ-algebra generated by the inverse images of cylinder sets. See also References External links Sigma Algebra from PlanetMath. Boolean algebra Experiment (probability theory) Families of sets Measure theory