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:
“What will our children remember of us, ten, fifteen years from now? The mobile we bought or didn’t buy? Or the tone in our voices, the look in our eyes, the enthusiasm for life—and for them—that we felt? They, and we, will remember the spirit of things, not the letter. Those memories will go so deep that no one could measure it, capture it, bronze it, or put it in a scrapbook.”
—Sonia Taitz (20th century)
“Thus the theory of description matters most.
It is the theory of the word for those
For whom the word is the making of the world,
The buzzing world and lisping firmament.”
—Wallace Stevens (1879–1955)
“...I’m thirteen years old, and I think I’m at the crossroads of my life. I’ve got to make good between now and the time I’m twenty, and I have only seven years to do it in. Besides, I’m the father of my family and I’ve got to earn all the money I can.”
—Mary Pickford (1893–1979)
“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)