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.

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