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:

    “Neither I nor anyone else knows what a standard is. We all recognize a dishonorable act, but have no idea what honor is.”
    —Anton Pavlovich Chekhov (1860–1904)

    “Truth is that concordance of an abstract statement with the ideal limit towards which endless investigation would tend to bring scientific belief, which concordance the abstract statement may possess by virtue of the confession of its inaccuracy and one-sidedness, and this confession is an essential ingredient of truth.”
    —Charles Sanders Peirce (1839–1914)