Search results
Results from the WOW.Com Content Network
Continuous functions, monotone functions, step functions, semicontinuous functions, Riemann-integrable functions, and functions of bounded variation are all Lebesgue measurable. [2] A function f : X → C {\displaystyle f:X\to \mathbb {C} } is measurable if and only if the real and imaginary parts are measurable.
In mathematics, a square-integrable function, also called a quadratically integrable function or function or square-summable function, [1] is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite.
The classical definition of a locally integrable function involves only measure theoretic and topological [4] concepts and can be carried over abstract to complex-valued functions on a topological measure space (X, Σ, μ): [5] however, since the most common application of such functions is to distribution theory on Euclidean spaces, [2] all ...
The eukaryotic cell cycle consists of four distinct phases: G 1 phase, S phase (synthesis), G 2 phase (collectively known as interphase) and M phase (mitosis and cytokinesis). M phase is itself composed of two tightly coupled processes: mitosis, in which the cell's nucleus divides, and cytokinesis, in which the cell's cytoplasm and cell membrane divides forming two daughter cells.
The term random function is also used to refer to a stochastic or random process, [25] [26] because a stochastic process can also be interpreted as a random element in a function space. [27] [28] The terms stochastic process and random process are used interchangeably, often with no specific mathematical space for the set that indexes the ...
A progressively measurable process, while defined quite technically, is important because it implies the stopped process is measurable. Being progressively measurable is a strictly stronger property than the notion of being an adapted process. [1] Progressively measurable processes are important in the theory of Itô integrals.
The integral of a non-negative general measurable function is then defined as an appropriate supremum of approximations by simple functions, and the integral of a (not necessarily positive) measurable function is the difference of two integrals of non-negative measurable functions. [1]
Consider a stochastic process X : [0, T] × Ω → R, and equip the real line R with its usual Borel sigma algebra generated by the open sets.. If we take the natural filtration F • X, where F t X is the σ-algebra generated by the pre-images X s −1 (B) for Borel subsets B of R and times 0 ≤ s ≤ t, then X is automatically F • X-adapted.