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:
“Rejoice, and men will seek you;
Grieve, and they turn and go.
They want full measure of all your pleasure,
But they do not need your woe”
—Ella Wheeler Wilcox (1850–1919)
“The weakness of the man who, when his theory works out into a flagrant contradiction of the facts, concludes “So much the worse for the facts: let them be altered,” instead of “So much the worse for my theory.””
—George Bernard Shaw (1856–1950)
“I know not how,
But I do find it cowardly and vile,
For fear of what might fall, so to prevent
The time of life—arming myself with patience
To stay the providence of some high powers
That govern us below.”
—William Shakespeare (1564–1616)
“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 (1874–1963)