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:
“Nobody is glad in the gladness of another, and our system is one of war, of an injurious superiority. Every child of the Saxon race is educated to wish to be first. It is our system; and a man comes to measure his greatness by the regrets, envies, and hatreds of his competitors.”
—Ralph Waldo Emerson (18031882)
“No theory is good unless it permits, not rest, but the greatest work. No theory is good except on condition that one use it to go on beyond.”
—André Gide (18691951)
“Everyone nowadays lives through too much and thinks through too little: they have a ravenous appetite and colic at the same time so that they keep getting thinner and thinner no matter how much they eat.Whoever says nowadays, I have not experienced anythingis a simpleton.”
—Friedrich Nietzsche (18441900)
“For these have governed in our lives,
And see how men have warred.
The Cross, the Crown, the Scales may all
As well have been the Sword.”
—Robert Frost (18741963)