Relation Between The Two Notions of Absolute Continuity
A finite measure μ on Borel subsets of the real line is absolutely continuous with respect to Lebesgue measure if and only if the point function
is locally an absolutely continuous real function. In other words, a function is locally absolutely continuous if and only if its distributional derivative is a measure that is absolutely continuous with respect to the Lebesgue measure.
If the absolute continuity holds then the Radon-Nikodym derivative of μ is equal almost everywhere to the derivative of F.
More generally, the measure μ is assumed to be locally finite (rather than finite) and F(x) is defined as μ((0,x]) for x>0, 0 for x=0, and -μ((x,0]) for x<0. In this case μ is the Lebesgue-Stieltjes measure generated by F. The relation between the two notions of absolute continuity still holds.
Read more about this topic: Absolute Continuity
Famous quotes containing the words relation, notions, absolute and/or continuity:
“Concord is just as idiotic as ever in relation to the spirits and their knockings. Most people here believe in a spiritual world ... in spirits which the very bullfrogs in our meadows would blackball. Their evil genius is seeing how low it can degrade them. The hooting of owls, the croaking of frogs, is celestial wisdom in comparison.”
—Henry David Thoreau (1817–1862)
“Even the simple act that we call “going to visit a person of our acquaintance” is in part an intellectual act. We fill the physical appearance of the person we see with all the notions we have about him, and in the totality of our impressions about him, these notions play the most important role.”
—Marcel Proust (1871–1922)
“I began revolution with 82 men. If I had [to] do it again, I do it with 10 or 15 and absolute faith. It does not matter how small you are if you have faith and plan of action.”
—Fidel Castro (b. 1926)
“Only the family, society’s smallest unit, can change and yet maintain enough continuity to rear children who will not be “strangers in a strange land,” who will be rooted firmly enough to grow and adapt.”
—Salvador Minuchin (20th century)