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:
“It is so manifestly incompatible with those precautions for our peace and safety, which all the great powers habitually observe and enforce in matters affecting them, that a shorter water way between our eastern and western seaboards should be dominated by any European government, that we may confidently expect that such a purpose will not be entertained by any friendly power.”
—Benjamin Harrison (1833–1901)