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In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, Fatou's lemma establishes an
inequality
Inequality may refer to:
Economics
* Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy
* Economic inequality, difference in economic well-being between population groups
* ...
relating the
Lebesgue integral
In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Lebe ...
of the
limit inferior
In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For ...
of a
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
of
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
s to the limit inferior of integrals of these functions. The
lemma is named after
Pierre Fatou
Pierre Joseph Louis Fatou (28 February 1878 – 9 August 1929) was a French mathematician and astronomer. He is known for major contributions to several branches of analysis. The Fatou lemma and the Fatou set are named after him.
Biography
P ...
.
Fatou's lemma can be used to prove the
Fatou–Lebesgue 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 ...
and Lebesgue's
dominated convergence theorem
In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which almost everywhere convergence of a sequence of functions implies convergence in the ''L''1 norm. Its power and utility are two of the primary ...
.
Standard statement
In what follows,
denotes the
-algebra of
Borel sets
In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are named ...
on
This_sequence_converges_uniformly_on_''S''_to_the_zero_function_and_the_limit,_0,_is_reached_in_a_finite_number_of_steps:_for_every_''x'' ≥ 0,_if_,_then_''f
n''(''x'') = 0.__However,_every_function_''f
n''_has_integral_−1.__Contrary_to_Fatou's_lemma,_this_value_is_strictly_less_than_the_integral_of_the_limit_(0).__
As_discussed_in__below,_the_problem_is_that_there_is_no_uniform_integrable_bound_on_the_sequence_from_below,_while_0_is_the_uniform_bound_from_above.
_Reverse_Fatou_lemma
Let_''f''
1,_''f''
2,_. . ._be_a_sequence_of_extended_real_number_line">extended_real
_
In__mathematics,_the_affinely_extended_real_number_system_is_obtained_from_the_real_number_system_\R_by_adding_two__infinity_elements:_+\infty_and_-\infty,_where_the_infinities_are_treated_as_actual_numbers._It_is_useful_in_describing_the_algebra__...
-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
:
\limsup_\int_S_f_n\,d\mu\leq\int_S\limsup_f_n\,d\mu.
Note:_Here_''g integrable''_means_that_''g''_is_measurable_and_that_
\textstyle\int_S_g\,d\mu<\infty.
_Sketch_of_proof
We_apply_linearity_of_Lebesgue_integral_and_Fatou's_lemma_to_the_sequence_
g_-_f_n.__Since_
\textstyle\int_Sg\,d\mu_<_+\infty,_this_sequence_is_defined_
\mu-almost_everywhere_and_non-negative.
_Extensions_and_variations_of_Fatou's_lemma
_Integrable_lower_bound
Let_''f''
1,_''f''
2,_. . ._be_a_sequence_of_extended_real-valued_measurable_functions_defined_on_a_measure_space_(''S'',''Σ'',''μ'')._If_there_exists_an_integrable_function_''g''_on_''S''_such_that_''f''
''n'' ≥ −''g''_for_all_''n'',_then
:
\int_S_\liminf__f_n\,d\mu
_\le_\liminf__\int_S_f_n\,d\mu.
_Proof
Apply_Fatou's_lemma_to_the_non-negative_sequence_given_by_''f''
''n'' + ''g''.
_Pointwise_convergence
If_in_the_previous_setting_the_sequence_''f''
1,_''f''
2,_. . ._Pointwise_convergence.html" ;"title="extended_real_number_line.html" "title=",2n.html" ;"title=",∞) with the Borel σ-algebra and the Lebesgue measure. For every natural number ''n'' define
:
f_n(x)=\begin-\frac1n&\textx\in [n,2n">,∞) with the Borel σ-algebra and the Lebesgue measure. For every natural number ''n'' define
:
f_n(x)=\begin-\frac1n&\textx\in [n,2n\\
0&\text
\end
This sequence converges uniformly on ''S'' to the zero function and the limit, 0, is reached in a finite number of steps: for every ''x'' ≥ 0, if , then ''fn''(''x'') = 0. However, every function ''fn'' has integral −1. Contrary to Fatou's lemma, this value is strictly less than the integral of the limit (0).
As discussed in below, the problem is that there is no uniform integrable bound on the sequence from below, while 0 is the uniform bound from above.
Reverse Fatou lemma
Let ''f''1, ''f''2, . . . be a sequence of extended real number line">extended real
In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra ...
-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''