Measure Theory On Time Scales
Associated with every time scale is a natural measure defined via
where denotes Lebesgue measure and is the backward shift operator defined on . The delta integral turns out to be the usual Lebesgue–Stieltjes integral with respect to this measure
and the delta derivative turns out to be the Radon–Nikodym derivative with respect to this measure
Read more about this topic: Time-scale Calculus
Famous quotes containing the words measure, theory, time and/or scales:
“The measure discriminates definitely against products which make up what has been universally considered a program of safe farming. The bill upholds as ideals of American farming the men who grow cotton, corn, rice, swine, tobacco, or wheat and nothing else. These are to be given special favors at the expense of the farmer who has toiled for years to build up a constructive farming enterprise to include a variety of crops and livestock.”
—Calvin Coolidge (1872–1933)
“... liberal intellectuals ... tend to have a classical theory of politics, in which the state has a monopoly of power; hoping that those in positions of authority may prove to be enlightened men, wielding power justly, they are natural, if cautious, allies of the “establishment.””
—Susan Sontag (b. 1933)
“Every true man is a cause, a country, and an age; requires infinite spaces and numbers and time fully to accomplish his design;—and posterity seem to follow his steps as a train of clients.”
—Ralph Waldo Emerson (1803–1882)
“As deaths have accumulated I have begun to think of life and death as a set of balance scales. When one is young, the scale is heavily tipped toward the living. With the first death, the first consciousness of death, the counter scale begins to fall. Death by death, the scales shift weight until what was unthinkable becomes merely a matter of gravity and the fall into death becomes an easy step.”
—Alison Hawthorne Deming (b. 1946)