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Φ ⊂ L 1 ( μ ) {\displaystyle \Phi \subset L^{1}(\mu )} is called uniformly integrable if sup f ∈ Φ ‖ f ‖ L 1 ( μ ) < ∞ {\displaystyle \sup _{f\in \Phi...

{\displaystyle X_{n}^{-}:=\max\{-X_{n},0\},\qquad n\in {\mathbb {N} },} are uniformly integrable with respect to the conditional expectation, in the sense that, for...

statement g {\displaystyle g} is integrable" means that the measurable function g {\displaystyle g} is Lebesgue integrable; i.e since g ≥ 0 {\displaystyle...

{\displaystyle {\mathcal {F}}\subset L^{1}(X,{\mathcal {A}},\mu )} is called uniformly integrable if lim M → + ∞ sup f ∈ F ∫ { | f | > M } | f | d μ = 0 {\displaystyle...

the limit exists then the function is said to be integrable (or more specifically Riemann-integrable). The Riemann sum can be made as close as desired...

{\displaystyle \{Z_{T}\mid T{\text{ a finite-valued stopping time}}\}} is uniformly integrable. Let Z {\displaystyle Z} be a cadlag supermartingale of class D....

the uniform limit of uniformly continuous functions is uniformly continuous; for a locally compact space, continuity is equivalent to local uniform continuity...

this would only be the case if the Radon–Nikodym derivative were a uniformly integrable martingale, which the exponential martingale described above is not...

_{n}X_{n}]} holds when the sequence { X n } {\displaystyle \{X_{n}\}} is uniformly integrable. The probability density function f X {\displaystyle f_{X}} of a...

then f is Riemann-integrable in [a, b]. A sequence (a1, a2, a3, ...) of real numbers is said to be equidistributed modulo 1 or uniformly distributed modulo...

} The function is Lebesgue integrable if and only if its absolute value is Lebesgue integrable (see Absolutely integrable function). Consider the indicator...

X_{n}{\xrightarrow {\overset {}{P}}}X} and the sequence (Xn) is uniformly integrable. If X n   → p   X {\displaystyle X_{n}\ \xrightarrow {\overset {}{p}}...

(\exp(M_{t}/2))_{t\geq 0}} is a uniformly integrable submartingale, then the Doléans-Dade exponential Ɛ(M) of M is a uniformly integrable martingale. Revuz, Daniel;...

following are equivalent: ( N t ) t > 0 {\displaystyle (N_{t})_{t>0}} is uniformly integrable, i.e. lim C → ∞ sup t > 0 ∫ { ω ∈ Ω ∣ | N t ( ω ) | > C } | N t (...

functions over a compact subset of the real line: absolutely continuous ⊆ uniformly continuous = {\displaystyle =} continuous and, for a compact interval...

be uniformly continuous so f {\displaystyle f} is not uniformly continuous. Any absolutely continuous function (over a compact interval) is uniformly continuous...

predictable process A are unbounded simple random walks (and not uniformly integrable), and Doob's optional stopping theorem might not be applicable to...

the entrepreneur has n machines, each having an operational lifetime uniformly distributed between zero and two years. Eric may let each machine run...

stochastic integration implies that H is X-integrable, with integral H · X = Y, if and only if Y0 = 0 and K · Y = (KH) · X. The set of X-integrable processes...

is the Arzelà–Ascoli theorem. Other cases of interest relate to uniform integrability, and the concept of normal family in complex analysis. Mahler's...