Symmetric Difference On Measure Spaces
As long as there is a notion of "how big" a set is, the symmetric difference between two sets can be considered a measure of how "far apart" they are. Formally, if μ is a σ-finite measure defined on a σ-algebra Σ, the function,
is a pseudometric on Σ. d becomes a metric if Σ is considered modulo the equivalence relation X ~ Y if and only if . The resulting metric space is separable if and only if L2(μ) is separable.
Read more about this topic: Symmetric Difference
Famous quotes containing the words difference, measure and/or spaces:
“Unhappiness is best defined as the difference between our talents and our expectations.”
—Edward De Bono (b. 1933)
“What we know partakes in no small measure of the nature of what has so happily been called the unutterable or ineffable, so that any attempt to utter or eff it is doomed to fail, doomed, doomed to fail.”
—Samuel Beckett (1906–1989)
“Though there were numerous vessels at this great distance in the horizon on every side, yet the vast spaces between them, like the spaces between the stars,—far as they were distant from us, so were they from one another,—nay, some were twice as far from each other as from us,—impressed us with a sense of the immensity of the ocean, the “unfruitful ocean,” as it has been called, and we could see what proportion man and his works bear to the globe.”
—Henry David Thoreau (1817–1862)