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)
“Don’t confuse hypothesis and theory. The former is a possible explanation; the latter, the correct one. The establishment of theory is the very purpose of science.”
—Martin H. Fischer (1879–1962)
“Today is made of yesterday, each time I steal
toward rites I do not know, waiting for the lost
ingredient, as if salt or money or even lust
would keep us calm and prove us whole at last.”
—Anne Sexton (1928–1974)
“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)