Uniform integrability

In mathematics, uniform integrability is an important concept in real analysis, functional analysis and measure theory, and plays a vital role in the theory of martingales. The definition used in measure theory is closely related to, but not identical to, the definition typically used in probability. In mathematics, uniform integrability is an important concept in real analysis, functional analysis and measure theory, and plays a vital role in the theory of martingales. The definition used in measure theory is closely related to, but not identical to, the definition typically used in probability. Textbooks on real analysis and measure theory often use the following definition. Let ( X , M , μ ) {displaystyle (X,{mathfrak {M}},mu )} be a positive measure space. A set Φ ⊂ L 1 ( μ ) {displaystyle Phi subset L^{1}(mu )} is called uniformly integrable if to each ε > 0 {displaystyle varepsilon >0} there corresponds a δ > 0 {displaystyle delta >0} such that whenever f ∈ Φ {displaystyle fin Phi } and μ ( E ) < δ . {displaystyle mu (E)<delta .} In the theory of probability, the following definition applies. The two probabilistic definitions are equivalent. The two definitions are closely related. A probability space is a measure space with total measure 1. A random variable is a real-valued measurable function on this space, and the expectation of a random variable is defined as the integral of this function with respect to the probability measure. Specifically, Let ( Ω , F , P ) {displaystyle (Omega ,{mathcal {F}},P)} be a probability space. Let the random variable X {displaystyle X} be a real-valued F {displaystyle {mathcal {F}}} -measurable function. Then the expectation of X {displaystyle X} is defined by