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:
“I realized early on that the academy and the literary world alike—and I don’t think there really is a distinction between the two—are always dominated by fools, knaves, charlatans and bureaucrats. And that being the case, any human being, male or female, of whatever status, who has a voice of her or his own, is not going to be liked.”
—Harold Bloom (b. 1930)