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The natural "Lebesgue measure" on S 1 is then the push-forward measure f ∗ (λ). The measure f ∗ (λ) might also be called "arc length measure" or "angle measure", since the f ∗ (λ)-measure of an arc in S 1 is precisely its arc length (or, equivalently, the angle that it subtends at the centre of the circle.)
Let : be a smooth map of smooth manifolds. Given , the differential of at is a linear map : from the tangent space of at to the tangent space of at (). The image of a tangent vector under is sometimes called the pushforward of by .
In finance, a T-forward measure is a pricing measure absolutely continuous with respect to a risk-neutral measure, but rather than using the money market as numeraire, it uses a bond with maturity T. The use of the forward measure was pioneered by Farshid Jamshidian (1987), and later used as a means of calculating the price of options on bonds .
The definition of the finite-dimensional distributions of a process is related to the definition for a measure in the following way: recall that the law of is a measure on the collection of all functions from into . In general, this is an infinite-dimensional space.
Pushforward measure, measure induced on the target measure space by a measurable function; Pushout (category theory), the categorical dual of pullback; Direct image sheaf, the pushforward of a sheaf by a map; Fiberwise integral, the direct image of a differential form or cohomology by a smooth map, defined by "integration on the fibres"
X is a Brownian motion with respect to P, i.e., the law of X with respect to P is the same as the law of an n-dimensional Brownian motion, i.e., the push-forward measure X ∗ (P) is classical Wiener measure on C 0 ([0, ∞); R n). both X is a martingale with respect to P (and its own natural filtration); and
In measure theory, a radonifying function (ultimately named after Johann Radon) between measurable spaces is one that takes a cylinder set measure (CSM) on the first space to a true measure on the second space. It acquired its name because the pushforward measure on the second space was historically thought of as a Radon measure.
A Borel measure on a separable Banach space is said to be a non-degenerate (centered) Gaussian measure if, for every linear functional except =, the push-forward measure is a non-degenerate (centered) Gaussian measure on in the sense defined above.