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 of action is the sentiment from which it proceeds. The greatest action may easily be one of the most private circumstance.”
—Ralph Waldo Emerson (1803–1882)
“The whole theory of modern education is radically unsound. Fortunately in England, at any rate, education produces no effect whatsoever. If it did, it would prove a serious danger to the upper classes, and probably lead to acts of violence in Grosvenor Square.”
—Oscar Wilde (1854–1900)
“I feel free as a bird. I’m in a unique position because I’m the boss. I buy what I like. I initiate things. I can experiment with all kinds of things I think the kids might be interested in. Nobody interferes. For me, it’s no chore to go to work. Most people never get to do this at any time in their lives.”
—Sarah Houghton, U.S. librarian. As quoted in Working, book 9, by Studs Terkel (1973)
“It cannot but affect our philosophy favorably to be reminded of these shoals of migratory fishes, of salmon, shad, alewives, marsh-bankers, and others, which penetrate up the innumerable rivers of our coast in the spring, even to the interior lakes, their scales gleaming in the sun; and again, of the fry which in still greater numbers wend their way downward to the sea.”
—Henry David Thoreau (1817–1862)