Dominated Convergence in -spaces (corollary)
Let be a measure space, a real number and a sequence of-measurable functions .
Assume the sequence converges -almost everywhere to an -measurable function, and is dominated by a, i.e., for every holds -almost everywhere.
Then all as well as are in and the sequence converges to in the sense of, i.e.: .
Idea of the proof: Apply the original theorem to the function sequence with the dominating function .
Read more about this topic: Dominated Convergence Theorem
Famous quotes containing the word dominated:
“In the middle of the next century, when the literary establishment will reflect the multicultural makeup of this country and not be dominated by assimiliationists with similar tastes, from similar backgrounds, and of similar pretensions, Langston Hughes will be to the twentieth century what Walt Whitman was to the nineteenth.”
—Ishmael Reed (b. 1938)