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Fatou–Lebesgue theorem

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Theorem in measure theory
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In mathematics, the Fatou–Lebesgue theorem establishes a chain of inequalities relating the integrals (in the sense of Lebesgue) of the limit inferior and the limit superior of a sequence of functions to the limit inferior and the limit superior of integrals of these functions. The theorem is named after Pierre Fatou and Henri Léon Lebesgue.

If the sequence of functions converges pointwise, the inequalities turn into equalities and the theorem reduces to Lebesgue's dominated convergence theorem.

Statement of the theorem

Let f1, f2, ... denote a sequence of real-valued measurable functions defined on a measure space (S,Σ,μ). If there exists a Lebesgue-integrable function g on S which dominates the sequence in absolute value, meaning that |fn| ≤ g for all natural numbers n, then all fn as well as the limit inferior and the limit superior of the fn are integrable and

∫ S lim inf n → ∞ f n d μ ≤ lim inf n → ∞ ∫ S f n d μ ≤ lim sup n → ∞ ∫ S f n d μ ≤ ∫ S lim sup n → ∞ f n d μ . {\displaystyle \int _{S}\liminf _{n\to \infty }f_{n}\,d\mu \leq \liminf _{n\to \infty }\int _{S}f_{n}\,d\mu \leq \limsup _{n\to \infty }\int _{S}f_{n}\,d\mu \leq \int _{S}\limsup _{n\to \infty }f_{n}\,d\mu \,.}

Here the limit inferior and the limit superior of the fn are taken pointwise. The integral of the absolute value of these limiting functions is bounded above by the integral of g.

Since the middle inequality (for sequences of real numbers) is always true, the directions of the other inequalities are easy to remember.

Proof

All fn as well as the limit inferior and the limit superior of the fn are measurable and dominated in absolute value by g, hence integrable.

Using linearity of the Lebesgue integral and applying Fatou's lemma to the non-negative functions f n + g {\displaystyle f_{n}+g} we get ∫ X lim inf n → ∞ f n d μ + ∫ X g d μ = ∫ X lim inf n → ∞ ( f n + g ) ≤ lim inf n → ∞ ∫ X ( f n + g ) d μ = lim inf n → ∞ ∫ X f n d μ + ∫ X g d μ . {\displaystyle \int _{X}\liminf _{n\to \infty }f_{n}\,d\mu +\int _{X}g\,d\mu =\int _{X}\liminf _{n\to \infty }(f_{n}+g)\leq \liminf _{n\to \infty }\int _{X}(f_{n}+g)\,d\mu =\liminf _{n\to \infty }\int _{X}f_{n}\,d\mu +\int _{X}g\,d\mu .} Cancelling the finite(!) ∫ X g d μ {\displaystyle \int _{X}g\,d\mu } term we get the first inequality. The second inequality is the elementary inequality between lim inf {\displaystyle \liminf } and lim sup {\displaystyle \limsup } . The last inequality follows by applying reverse Fatou lemma, i.e. applying the Fatou lemma to the non-negative functions g − f n {\displaystyle g-f_{n}} , and again (up to sign) cancelling the finite ∫ X g d μ {\displaystyle \int _{X}g\,d\mu } term.

Finally, since lim sup n | f n | ≤ g {\displaystyle \limsup _{n}|f_{n}|\leq g} ,

max ( | ∫ S lim inf n → ∞ f n d μ | , | ∫ S lim sup n → ∞ f n d μ | ) ≤ ∫ S max ( | lim inf n → ∞ f n | , | lim sup n → ∞ f n | ) d μ ≤ ∫ S lim sup n → ∞ | f n | d μ ≤ ∫ S g d μ {\displaystyle \max {\bigl (}{\biggl |}\int _{S}\liminf _{n\to \infty }f_{n}\,d\mu {\biggr |},{\Bigl |}\int _{S}\limsup _{n\to \infty }f_{n}\,d\mu {\Bigr |}{\bigr )}\leq \int _{S}\max {\bigl (}{\Bigl |}\liminf _{n\to \infty }f_{n}{\Bigr |},{\Bigl |}\limsup _{n\to \infty }f_{n}{\Bigr |}{\bigr )}\,d\mu \leq \int _{S}\limsup _{n\to \infty }|f_{n}|\,d\mu \leq \int _{S}g\,d\mu }

by the monotonicity of the Lebesgue integral.

References

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