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:
“Common sense is the measure of the possible; it is composed of experience and prevision; it is calculation appled to life.”
—Henri-Frédéric Amiel (1821–1881)
“... the first reason for psychology’s failure to understand what people are and how they act, is that clinicians and psychiatrists, who are generally the theoreticians on these matters, have essentially made up myths without any evidence to support them; the second reason for psychology’s failure is that personality theory has looked for inner traits when it should have been looking for social context.”
—Naomi Weisstein (b. 1939)
“O Time and Change!—with hair as gray
As was my sire’s that winter day,
How strange it seems, with so much gone
Of life and love, to still live on!
Ah, brother! only I and thou
Are left of all that circle now,—
The dear home faces whereupon
That fitful firelight paled and shone.”
—John Greenleaf Whittier (1807–1892)
“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)