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 between, relation, notions, absolute and/or continuity:
“You know there are no secrets in America. It’s quite different in England, where people think of a secret as a shared relation between two people.”
—W.H. (Wystan Hugh)
“The instincts of the ant are very unimportant, considered as the ant’s; but the moment a ray of relation is seen to extend from it to man, and the little drudge is seen to be a monitor, a little body with a mighty heart, then all its habits, even that said to be recently observed, that it never sleeps, become sublime.”
—Ralph Waldo Emerson (1803–1882)
“Shelley is truth itself—and honour itself—notwithstanding his out-of-the-way notions about religion.”
—George Gordon Noel Byron (1788–1824)
“The teacher must derive not only the capacity, but the desire, to observe natural phenomena. In our system, she must become a passive, much more than an active, influence, and her passivity shall be composed of anxious scientific curiosity and of absolute respect for the phenomenon which she wishes to observe. The teacher must understand and feel her position of observer: the activity must lie in the phenomenon.”
—Maria Montessori (1870–1952)
“If you associate enough with older people who do enjoy their lives, who are not stored away in any golden ghettos, you will gain a sense of continuity and of the possibility for a full life.”
—Margaret Mead (1901–1978)