In mathematics, more specifically measure theory, there are various notions of the convergence of measures. For an intuitive general sense of what is meant by convergence in measure, consider a sequence of measures on a space, sharing a common collection of measurable sets. Such a sequence might represent an attempt to construct 'better and better' approximations to a desired measure that is difficult to obtain directly. The meaning of 'better and better' is subject to all the usual caveats for taking limits; for any error tolerance we require there be sufficiently large for to ensure the 'difference' between and is smaller than . Various notions of convergence specify precisely what the word 'difference' should mean in that description; these notions are not equivalent to one another, and vary in strength.
Three of the most common notions of convergence are described below.
Read more about Convergence Of Measures: Informal Descriptions, Total Variation Convergence of Measures, Strong Convergence of Measures, Weak Convergence of Measures
Famous quotes containing the word measures:
“Him the Almighty Power
Hurld headlong flaming from th’ Ethereal Skie
With hideous ruine and combustion down
To bottomless perdition, there to dwell
In Adamantine Chains and penal Fire,
Who durst defie th’ Omnipotent to Arms.
Nine times the Space that measures Day and Night
To mortal men, he with his horrid crew
Lay vanquisht, rowling in the fiery Gulfe”
—John Milton (1608–1674)