In mathematics, Fatou's lemma establishes an inequality relating the integral (in the sense of Lebesgue) of the limit inferior of a sequence of functions to the limit inferior of integrals of these functions. The lemma is named after the French mathematician Pierre Fatou (1878 - 1929). For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... This article is about inequalities in mathematics. ... This article is about the concept of integrals in calculus. ... The integral of a positive function can be interpreted as the area under a curve. ... In mathematics, the limit inferior and limit superior (also called infimum limit and supremum limit) of a sequence can be thought of as limiting bounds on the sequence. ... For other senses of this word, see sequence (disambiguation). ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... In mathematics, a lemma is a proven proposition which is used as a stepping stone to a larger result rather than an independent statement, in and of itself. ... Leonhard Euler, considered one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ... Pierre Fatou was the first to define the Mandelbrot set. ... 1878 (MDCCCLXXVIII) was a common year starting on Tuesday (see link for calendar). ... Year 1929 (MCMXXIX) was a common year starting on Tuesday (link will display the full calendar) of the Gregorian calendar. ...

Fatou's lemma can be used to prove the Fatou–Lebesgue theorem and Lebesgue's dominated convergence theorem. 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. ... In mathematics, Lebesgues dominated convergence theorem states that if a sequence { fn : n = 1, 2, 3, ... } of real-valued measurable functions on a measure space S converges almost everywhere, and is dominated (explained below) by some nonnegative function g in , then It is proven using Fatous lemma. ...

Standard statement of Fatou's lemma

If f₁, f₂, . . . is a sequence of non-negative measurable functions defined on a measure space (S,Σ,μ), then A negative number is a number that is less than zero, such as −3. ... In mathematics, measurable functions are well-behaved functions between measurable spaces. ... In mathematics, a measure is a function that assigns a number, e. ...

Note: On the left-hand side the limit inferior of the f_n is taken pointwise. The functions are allowed to attain the value infinity and the integrals may also be infinite. Suppose { fn } is a sequence of functions sharing the same domain in common (for the moment, we defer making precise the nature of the values of these functions, but the reader may take them to be real numbers if that makes anyone feel good). ... For other uses, see Infinity (disambiguation). ...

Proof

Fatou's lemma is here proved using the monotone convergence theorem (it can be proved directly). Monotone convergence theorem, in mathematics, may refer to several theorems, all of which are concerned with a monotonic function in one way or another: Monotonic function refers to the convergence of an infinite series that is monotonic Dominated convergence theorem refers to Lebesgues monotone convergence theorem Categories: | | ...

Let f denote the limit inferior of the f_n. For every natural number k define pointwise the function

Then the sequence g₁, g₂, . . . is increasing and converges pointwise to f. For k ≤ n, we have g_k ≤ f_n, so that

hence

Using the monotone convergence theorem, the last inequality, and the definition of the limit inferior, it follows that

Examples for strict inequality

Equip the space S with the Borel σ-algebra and the Lebesgue measure. In mathematics, the Borel algebra (or Borel σ-algebra) on a topological space X is a σ-algebra of subsets of X associated to the topology of X. In the mathematics literature, there are at least two inequivalent definitions of this σ-algebra: The minimal σ-algebra containing the open sets. ... In mathematics, the Lebesgue measure is the standard way of assigning a length, area or volume to subsets of Euclidean space. ...

• Example for a probability space: Let S = [0,1] denote the unit interval. For every natural number n define

• Example with uniform convergence: Let S denote the set of all real numbers. Define

These sequences converge on S pointwise (respectively uniform) to the zero function (with zero integral), but every f_n has integral one. In mathematics, the definition of the probability space is the foundation of probability theory. ... In mathematics, the unit interval is the interval [0,1], that is the set of all real numbers x such that zero is less than or equal to x and x is less than or equal to one. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... In the mathematical field of analysis, uniform convergence is a type of convergence stronger than pointwise convergence. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... Zero redirects here. ...

Reverse Fatou lemma

Let f₁, f₂, . . . be a sequence of extended real-valued measurable functions defined on a measure space (S,Σ,μ). If there exists a non-negative integrable function g on S such that f_n ≤ g for all n, then The extended real number line is obtained from the real number line R by adding two elements: +∞ and −∞ (which are not considered to be real numbers). ...

Note: Here g integrable means that g is measurable and that .

Proof

Apply Fatou's lemma to the non-negative sequence given by g – f_n.

Extensions and variations of Fatou's lemma

Integrable lower bound

Let f₁, f₂, . . . be a sequence of extended real-valued measurable functions defined on a measure space (S,Σ,μ). If there exists a non-negative integrable function g on S such that f_n ≥ −g for all n, then

Proof

Apply Fatou's lemma to the non-negative sequence given by f_n + g.

Pointwise convergence

If in the previous setting the sequence f₁, f₂, . . . converges pointwise to a function f μ-almost everywhere on S, then Suppose { fn } is a sequence of functions sharing the same domain in common (for the moment, we defer making precise the nature of the values of these functions, but the reader may take them to be real numbers if that makes anyone feel good). ... In measure theory (a branch of mathematical analysis), one says that a property holds almost everywhere if the set of elements for which the property does not hold is a null set, i. ...

Proof

Note that f has to agree with the limit inferior of the functions f_n almost everywhere, and that the values of the integrand on a set of measure zero have no influence on the value of the integral.

Convergence in measure

The last assertion also holds, if the sequence f₁, f₂, . . . converges in measure to a function f. In mathematics, particularly in measure theory, convergence in measure is a weak notion of convergence of measurable functions. ...

Proof

There exists a subsequence such that

Since this subsequence also converges in measure to f, there exists a further subsequence, which converges pointwise to f almost everywhere, hence the previous variation of Fatou's lemma is applicable to this subsubsequence.

Fatou's lemma for conditional expectations

In probability theory, by a change of notation, the above versions of Fatou's lemma are applicable to sequences of random variables X₁, X₂, . . . defined on a probability space ; the integrals turn into expectations. In addition, there is also a version for conditional expectations: Probability theory is the branch of mathematics concerned with analysis of random phenomena. ... A random variable can be thought of as the numeric result of operating a non-deterministic mechanism or performing a non-deterministic experiment to generate a random result. ... In mathematics, the definition of the probability space is the foundation of probability theory. ... In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are... In probability theory, a conditional expectation is the expected value of a real random variable with respect to a conditional probability distribution. ...

Let X₁, X₂, . . . be a sequence of non-negative random variables on a probability space and let be a sub-σ-algebra. Then In mathematics, a σ-algebra (pronounced sigma-algebra) or σ-field over a set X is a collection Σ of subsets of X that is closed under countable set operations, meaning that the union or the intersection of countably many members of the algebra is also a member. ...

   almost surely.

Note: Conditional expectation for non-negative random variables is always well defined, finite expectation is not needed. In probability theory, an event happens almost surely (a. ...

Proof

Besides a change of notation, the proof is very similar to the one for the standard version of Fatou's lemma above, however the monotone convergence theorem for conditional expectations has to be applied. Monotone convergence theorem, in mathematics, may refer to several theorems, all of which are concerned with a monotonic function in one way or another: Monotonic function refers to the convergence of an infinite series that is monotonic Dominated convergence theorem refers to Lebesgues monotone convergence theorem Categories: | | ...

Let X denote the limit inferior of the X_n. For every natural number k define pointwise the random variable

Then the sequence Y₁, Y₂, . . . is increasing and converges pointwise to X. For k ≤ n, we have Y_k ≤ X_n, so that

   almost surely

by the monotonicity of conditional expectation, hence In probability theory, a conditional expectation is the expected value of a real random variable with respect to a conditional probability distribution. ...

   almost surely,

because the countable union of the exceptional sets of probability zero is again a null set. Using the definition of X, its representation as pointwise limit of the Y_k, the monotone convergence theorem for conditional expectations, the last inequality, and the definition of the limit inferior, it follows that almost surely In measure theory, a null set is a set that is negligible for the purposes of the measure in question. ...

External links

• Fatou's lemma on PlanetMath

PlanetMath is a free, collaborative, online mathematics encyclopedia. ...

References

• H.L. Royden, "Real Analysis", Prentice Hall, 1988.