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 your quality as a public person, as a citizen, is the gap between what you do and what you say.”
—Ramsey Clark (b. 1927)
“Many people have an oversimplified picture of bonding that could be called the “epoxy” theory of relationships...if you don’t get properly “glued” to your babies at exactly the right time, which only occurs very soon after birth, then you will have missed your chance.”
—Pamela Patrick Novotny (20th century)
“Before I had my first child, I never really looked forward in anticipation to the future. As I watched my son grow and learn, I began to imagine the world this generation of children would live in. I thought of the children they would have, and of their children. I felt connected to life both before my time and beyond it. Children are our link to future generations that we will never see.”
—Louise Hart (20th century)
“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)