Fatou's Lemma - Standard Statement of Fatou's Lemma

Standard Statement of Fatou's Lemma

Let f1, f2, f3, . . . be a sequence of non-negative measurable functions on a measure space (S,Σ,μ). Define the function f : S → a.e. pointwise limit by

f(s) =\liminf_{n\to\infty} f_n(s),\qquad s\in S.

Then f  is measurable and

\int_S f\,d\mu \le \liminf_{n\to\infty} \int_S f_n\,d\mu\,.

Note: The functions are allowed to attain the value +∞ and the integrals may also be infinite.

Read more about this topic:  Fatou's Lemma

Famous quotes containing the words standard and/or statement:

    “Society’s double behavioral standard for women and for men is, in fact, a more effective deterrent than economic discrimination because it is more insidious, less tangible. Economic disadvantages involve ascertainable amounts, but the very nature of societal value judgments makes them harder to define, their effects harder to relate.”
    —Anne Tucker (b. 1945)

    “The honor my country shall never be stained by an apology from me for the statement of truth and the performance of duty; nor can I give any explanation of my official acts except such as is due to integrity and justice and consistent with the principles on which our institutions have been framed.”
    —Andrew Jackson (1767–1845)