# Fatou's lemma

In mathematics, Fatou's lemma establishes an inequality relating the Lebesgue integral of the limit inferior of a sequence of functions to the limit inferior of integrals of these functions. The lemma is named after Pierre Fatou.

Fatou's lemma can be used to prove the Fatou–Lebesgue theorem and Lebesgue's dominated convergence theorem.

## Standard statement of Fatou's lemma

In what follows, $\operatorname {\mathcal {B}} _{\mathbb {R} _{\geq 0}}$  denotes the $\sigma$ -algebra of Borel sets on $[0,+\infty ]$ .

Fatou's lemma. Given a measure space $(\Omega ,{\mathcal {F}},\mu )$  and a set $X\in {\mathcal {F}},$  let $\{f_{n}\}$  be a sequence of $({\mathcal {F}},\operatorname {\mathcal {B}} _{\mathbb {R} _{\geq 0}})$ -measurable non-negative functions $f_{n}:X\to [0,+\infty ]$ . Define the function $f:X\to [0,+\infty ]$  by setting

$f(x)=\liminf _{n\to \infty }f_{n}(x),$

for every $x\in X$ . Then $f$  is $({\mathcal {F}},\operatorname {\mathcal {B}} _{\mathbb {R} _{\geq 0}})$ -measurable, and

$\int _{X}f\,d\mu \leq \liminf _{n\to \infty }\int _{X}f_{n}\,d\mu .$

Remark 1. The integrals may be finite or infinite.

Remark 2. Fatou's lemma remains true if its assumptions hold $\mu$ -almost everywhere. In other words, it is enough that there is a null set $N$  such that the sequence $\{f_{n}(x)\}$  non-decreases for every ${x\in X\setminus N}.$  To see why this is true, we start with an observation that allowing the sequence $\{f_{n}\}$  to pointwise non-decrease almost everywhere causes its pointwise limit $f$  to be undefined on some null set $N$ . On that null set, $f$  may then be defined arbitrarily, e.g. as zero, or in any other way that preserves measurability. To see why this will not affect the outcome, note that since ${\mu (N)=0},$  we have, for every $k,$

$\int _{X}f_{k}\,d\mu =\int _{X\setminus N}f_{k}\,d\mu$  and $\int _{X}f\,d\mu =\int _{X\setminus N}f\,d\mu ,$

provided that $f$  is $({\mathcal {F}},\operatorname {\mathcal {B}} _{\mathbb {R} _{\geq 0}})$ -measurable. (These equalities follow directly from the definition of Lebesgue integral for a non-negative function).

For use in the proof, define a sequence of functions by $\textstyle g_{n}(x):=\inf _{k\geq n}f_{k}(x)$ .

Remark 3. For every $x\in X$ ,

1. The non-negative sequence $\{g_{n}(x)\}_{n}$  is non-decreasing, i.e. $g_{n}(x)\leq g_{n+1}(x)$ , for every $n\geq 1$ ;
2. By definition of limit inferior, $f(x)=\liminf _{n\to \infty }f_{n}(x):=\lim _{n\to \infty }\inf _{k\geq n}f_{k}(x)=\lim _{n\to \infty }g_{n}(x)$

Remark 4. The proof below does not use any properties of Lebesgue integral except those established here.

Remark 5 (monotonicity of Lebesgue integral). In the proof below, we apply the monotonic property of Lebesgue integral to non-negative functions only. Specifically (see Remark 4), let the functions $f,g:X\to [0,+\infty ]$  be $({\mathcal {F}},\operatorname {\mathcal {B}} _{\mathbb {R} _{\geq 0}})$ -measurable.

• If $f\leq g$  everywhere on $X,$  then
$\int _{X}f\,d\mu \leq \int _{X}g\,d\mu .$
• If $X_{1},X_{2}\in {\mathcal {F}}$  and $X_{1}\subseteq X_{2},$  then
$\int _{X_{1}}f\,d\mu \leq \int _{X_{2}}f\,d\mu .$

Proof. Denote $\operatorname {SF} (h)$  the set of simple $({\mathcal {F}},\operatorname {\mathcal {B}} _{\mathbb {R} _{\geq 0}})$ -measurable functions $s:X\to [0,\infty )$  such that $0\leq s\leq h$  everywhere on $X.$

1. Since $f\leq g,$  we have

$\operatorname {SF} (f)\subseteq \operatorname {SF} (g).$

By definition of Lebesgue integral and the properties of supremum,

$\int _{X}f\,d\mu =\sup _{s\in {\rm {SF}}(f)}\int _{X}s\,d\mu \leq \sup _{s\in {\rm {SF}}(g)}\int _{X}s\,d\mu =\int _{X}g\,d\mu .$

2. Let ${\mathbf {1} }_{X_{1}}$  be the indicator function of the set $X_{1}.$  It can be deduced from the definition of Lebesgue integral that

$\int _{X_{2}}f\cdot {\mathbf {1} }_{X_{1}}\,d\mu =\int _{X_{1}}f\,d\mu$

if we notice that, for every $s\in {\rm {SF}}(f\cdot {\mathbf {1} }_{X_{1}}),$  $s=0$  outside of $X_{1}.$  Combined with the previous property, the inequality $f\cdot {\mathbf {1} }_{X_{1}}\leq f$  implies

$\int _{X_{1}}f\,d\mu =\int _{X_{2}}f\cdot {\mathbf {1} }_{X_{1}}\,d\mu \leq \int _{X_{2}}f\,d\mu .$

### Proof

This proof does not rely on the monotone convergence theorem. However, we do explain how that theorem may be applied.

For those not interested in independent proof, the intermediate results below may be skipped.

#### Intermediate results

##### Lebesgue integral as measure

Lemma 1. Let $(\Omega ,{\mathcal {F}},\mu )$  be a measurable space. Consider a simple $({\mathcal {F}},\operatorname {\mathcal {B}} _{\mathbb {R} _{\geq 0}})$ -measurable non-negative function $s:\Omega \to {\mathbb {R} _{\geq 0}}$ . For a subset $S\subseteq \Omega$ , define

$\nu (S)=\int _{S}s\,d\mu$ .

Then $\nu$  is a measure on $\Omega$ .

###### Proof

We will only prove countable additivity, leaving the rest up to the reader. Let $S=\cup _{i=1}^{\infty }S_{i}$ , where all the sets $S_{i}$  are pairwise disjoint. Due to simplicity,

$s=\sum _{i=1}^{n}c_{i}\cdot {\mathbf {1} }_{A_{i}}$ ,

for some finite non-negative constants $c_{i}\in {\mathbb {R} }_{\geq 0}$  and pairwise disjoint sets $A_{i}\in {\mathcal {F}}$  such that $\cup _{i=1}^{n}A_{i}=\Omega$ . By definition of Lebesgue integral,

{\begin{aligned}\nu (S)&=\\&=\sum _{i=1}^{n}c_{i}\cdot \mu (S\cap A_{i})\\&=\sum _{i=1}^{n}c_{i}\cdot \mu {\bigl (}(\cup _{j=1}^{\infty }S_{j})\cap A_{i}{\bigr )}\\&=\sum _{i=1}^{n}c_{i}\cdot \mu {\bigl (}\cup _{j=1}^{\infty }(S_{j}\cap A_{i}){\bigr )}\end{aligned}}

Since all the sets $S_{j}\cap A_{i}$  are pairwise disjoint, the countable additivity of $\mu$  gives us

$\sum _{i=1}^{n}c_{i}\cdot \mu {\bigl (}\cup _{j=1}^{\infty }(S_{j}\cap A_{i}){\bigr )}=\sum _{i=1}^{n}c_{i}\cdot \sum _{j=1}^{\infty }\mu (S_{j}\cap A_{i}).$

Since all the summands are non-negative, the sum of the series, whether this sum is finite or infinite, cannot change if summation order does because the series is either absolutely convergent or diverges to $+\infty .$  For that reason,

{\begin{aligned}\sum _{i=1}^{n}c_{i}\cdot \sum _{j=1}^{\infty }\mu (S_{j}\cap A_{i})&=\sum _{j=1}^{\infty }\sum _{i=1}^{n}c_{i}\cdot \mu (S_{j}\cap A_{i})\\&=\sum _{j=1}^{\infty }\int _{S_{j}}s\,d\mu \\&=\sum _{j=1}^{\infty }\nu (S_{j}),\end{aligned}}

as required.

##### "Continuity from below"

The following property is a direct consequence of the definition of measure.

Lemma 2. Let $\mu$  be a measure, and $S=\cup _{i=1}^{\infty }S_{i}$ , where

$S_{1}\subseteq \ldots \subseteq S_{i}\subseteq S_{i+1}\subseteq \ldots \subseteq S$

is a non-decreasing chain with all its sets $\mu$ -measurable. Then

$\mu (S)=\lim _{i}\mu (S_{i})$ .

#### Proof of theorem

Step 1. $g_{n}=g_{n}(x)$  is $({\mathcal {F}},\operatorname {\mathcal {B}} _{\mathbb {R} _{\geq 0}})$ -measurable, for every $n\geq 1$ .

Indeed, since the Borel $\sigma$ -algebra on $\mathbb {R} \cup \{\pm \infty \}$  is generated by the closed intervals $\{[t,+\infty ]\}_{-\infty \leq t\leq +\infty }$ , it suffices to show that, $g_{n}^{-1}([t,+\infty ])\in {\mathcal {F}}$ , for every $t\in [-\infty ,+\infty ]$ , where $g_{n}^{-1}([t,+\infty ])$  denotes the inverse image of $[t,+\infty ]$  under $g_{n}$ .

Observe that

$g_{n}(x)\geq t\quad \Leftrightarrow \quad {\Bigl (}\forall k\geq n\quad f_{k}(x)\geq t{\Bigr )}$ ,

or equivalently,

{\begin{aligned}g_{n}^{-1}([t,+\infty ])&=\left\{x\in X\mid g_{n}(x)\geq t\right\}\\[3pt]&=\bigcap _{k}\left\{x\in X\mid f_{k}(x)\geq t\right\}\\[3pt]&=\bigcap _{k}f_{k}^{-1}([t,+\infty ])\end{aligned}}

Note that every set on the right-hand side is from ${\mathcal {F}}$ . Since, by definition, ${\mathcal {F}}$  is closed under countable intersections, we conclude that the left-hand side is also a member of ${\mathcal {F}}$ . The $({\mathcal {F}},\operatorname {\mathcal {B}} _{\mathbb {R} _{\geq 0}})$ -measurability of $g_{n}$  follows.

Step 2. Now, we want to show that the function $f$  is $({\mathcal {F}},\operatorname {\mathcal {B}} _{\mathbb {R} _{\geq 0}})$ -measurable.

If we were to use the monotone convergence theorem, the measurability of $f$  would follow easily from Remark 3.

Alternatively, using the technique from Step 1, it is enough to verify that $f^{-1}([0,t])\in {\mathcal {F}}$ , for every $t\in [-\infty ,+\infty ]$ . Since the sequence $\{g_{n}(x)\}$  pointwise non-decreases (see Remark 3), arguing as above, we get

$0\leq f(x)\leq t\quad \Leftrightarrow \quad {\Bigl (}\forall n\quad 0\leq g_{n}(x)\leq t{\Bigr )}$ .

Due to the measurability of $g_{n}$ , the above equivalency implies that

$f^{-1}([0,t])=\bigcap _{n}g_{n}^{-1}([0,t])\in {\mathcal {F}}$ .

End of Step 2.

The proof can proceed in two ways.

Proof using the monotone convergence theorem. By definition, $\textstyle g_{n}(x)=\inf _{k\geq n}f_{k}(x)$ , so we have $g_{n}\leq f_{n}$ , $\forall n\in \mathbb {N}$ , and furthermore the sequence $\{g_{n}(x)\}_{n}$ is non-decreasing $\forall x\in X$ . Recall that $f(x)=\liminf _{n\to \infty }f_{n}(x)=\lim _{n\to \infty }\inf _{k\geq n}f_{k}(x)$ , and therefore:

{\begin{aligned}\int _{X}f\,d\mu &=\int _{X}\lim _{n}g_{n}\,d\mu \\&=\lim _{n}\int _{X}g_{n}\,d\mu \\&=\liminf _{n}\int _{X}g_{n}\,d\mu \\&\leq \liminf _{n}\int _{X}f_{n}\,d\mu ,\end{aligned}}

as required.

Independent proof. To prove the inequality without using the monotone convergence theorem, we need some extra machinery. Denote $\operatorname {SF} (f)$  the set of simple $({\mathcal {F}},\operatorname {\mathcal {B}} _{\mathbb {R} _{\geq 0}})$ -measurable functions $s:X\to [0,\infty )$  such that $0\leq s\leq f$  on $X$ .

Step 3. Given a simple function $s\in \operatorname {SF} (f)$  and a real number $t\in (0,1)$ , define

$B_{k}^{s,t}=\{x\in X\mid t\cdot s(x)\leq g_{k}(x)\}\subseteq X.$

Then $B_{k}^{s,t}\in {\mathcal {F}}$ , $B_{k}^{s,t}\subseteq B_{k+1}^{s,t}$ , and $\textstyle X=\bigcup _{k}B_{k}^{s,t}$ .

Step 3a. To prove the first claim, let

$s=\sum _{i=1}^{m}c_{i}\cdot \mathbf {1} _{A_{i}},$

for some finite collection of pairwise disjoint measurable sets $A_{1},\ldots ,A_{m}\in {\mathcal {F}}$  such that $\textstyle X=\cup _{i=1}^{m}A_{i}$ , some (finite) real values $c_{1},\ldots ,c_{m}$ , and $\mathbf {1} _{A_{i}}$  denoting the indicator function of the set $A_{i}$ . Then

$B_{k}^{s,t}=\bigcup _{i=1}^{m}{\Bigl (}g_{k}^{-1}{\Bigl (}[t\cdot c_{i},+\infty ]{\Bigr )}\cap A_{i}{\Bigr )}$ .

Since the pre-image $g_{k}^{-1}{\Bigl (}[t\cdot c_{i},+\infty ]{\Bigr )}$  of the Borel set $[t\cdot c_{i},+\infty ]$  under the measurable function $g_{k}$  is measurable, and $\sigma$ -algebras, by definition, are closed under finite intersection and unions, the first claim follows.

Step 3b. To prove the second claim, note that, for each $k$  and every $x\in X$ , $g_{k}(x)\leq g_{k+1}(x).$

Step 3c. To prove the third claim, we show that $\textstyle X\subseteq \bigcup _{k}B_{k}^{s,t}$ .

Indeed, if, to the contrary, $\textstyle X\not \subseteq \bigcup _{k}B_{k}^{s,t}$ , then an element

$x_{0}\in X\setminus \bigcup _{k}B_{k}^{s,t}=\bigcap _{k}(X\setminus B_{k}^{s,t})$

exists such that $g_{k}(x_{0}) , for every $k$ . Taking the limit as $k\to \infty$ , get

$f(x_{0})\leq t\cdot s(x_{0})

But by initial assumption, $s\leq f$ . This is a contradiction.

Step 4. For every simple $({\mathcal {F}},\operatorname {\mathcal {B}} _{\mathbb {R} _{\geq 0}})$ -measurable non-negative function $s_{2}$ ,

$\lim _{n}\int _{B_{n}^{s,t}}s_{2}\,d\mu =\int _{X}s_{2}\,d\mu .$

To prove this, define $\textstyle \nu (S)=\int _{S}s_{2}\,d\mu$ . By Lemma 1, $\nu (S)$  is a measure on $\Omega$ . By "continuity from below" (Lemma 2),

$\lim _{n}\int _{B_{n}^{s,t}}s_{2}\,d\mu =\lim _{n}\nu (B_{n}^{s,t})=\nu (X)=\int _{X}s_{2}\,d\mu$ ,

as required.

Step 5. We now prove that, for every $s\in \operatorname {SF} (f)$ ,

$\int _{X}s\,d\mu \leq \lim _{k}\int _{X}g_{k}\,d\mu$ .

Indeed, using the definition of $B_{k}^{s,t}$ , the non-negativity of $g_{k}$ , and the monotonicity of Lebesgue integral, we have

$\forall k\geq 1\qquad \int _{B_{k}^{s,t}}t\cdot s\,d\mu \leq \int _{B_{k}^{s,t}}g_{k}\,d\mu \leq \int _{X}g_{k}\,d\mu$ .

In accordance with Step 4, as $k\to \infty$  the inequality becomes

$t\int _{X}s\,d\mu \leq \lim _{k}\int _{X}g_{k}\,d\mu$ .

Taking the limit as $t\uparrow 1$  yields

$\int _{X}s\,d\mu \leq \lim _{k}\int _{X}g_{k}\,d\mu$ ,

as required.

Step 6. To complete the proof, we apply the definition of Lebesgue integral to the inequality established in Step 5 and take into account that $g_{n}\leq f_{n}$ :

{\begin{aligned}\int _{X}f\,d\mu &=\sup _{s\in \operatorname {SF} (f)}\int _{X}s\,d\mu \\&\leq \lim _{k}\int _{X}g_{k}\,d\mu \\&=\liminf _{k}\int _{X}g_{k}\,d\mu \\&\leq \liminf _{k}\int _{X}f_{k}\,d\mu \end{aligned}}

The proof is complete.

## Examples for strict inequality

Equip the space $S$  with the Borel σ-algebra and the Lebesgue measure.

• Example for a probability space: Let $S=[0,1]$  denote the unit interval. For every natural number $n$  define
$f_{n}(x)={\begin{cases}n&{\text{for }}x\in (0,1/n),\\0&{\text{otherwise.}}\end{cases}}$
• Example with uniform convergence: Let $S$  denote the set of all real numbers. Define
$f_{n}(x)={\begin{cases}{\frac {1}{n}}&{\text{for }}x\in [0,n],\\0&{\text{otherwise.}}\end{cases}}$

These sequences $(f_{n})_{n\in \mathbb {N} }$  converge on $S$  pointwise (respectively uniformly) to the zero function (with zero integral), but every $f_{n}$  has integral one.

## The role of non-negativity

A suitable assumption concerning the negative parts of the sequence f1, f2, . . . of functions is necessary for Fatou's lemma, as the following example shows. Let S denote the half line [0,∞) with the Borel σ-algebra and the Lebesgue measure. For every natural number n define

$f_{n}(x)={\begin{cases}-{\frac {1}{n}}&{\text{for }}x\in [n,2n],\\0&{\text{otherwise.}}\end{cases}}$

This sequence converges uniformly on S to the zero function (with zero integral) and for every x ≥ 0 we even have fn(x) = 0 for all n > x (so for every point x the limit 0 is reached in a finite number of steps). However, every function fn has integral −1, hence the inequality in Fatou's lemma fails. As shown 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 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 fn ≤ g for all n, then

$\limsup _{n\to \infty }\int _{S}f_{n}\,d\mu \leq \int _{S}\limsup _{n\to \infty }f_{n}\,d\mu .$

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 $\textstyle \int _{S}g\,d\mu <+\infty ,$  this sequence is defined $\mu$ -almost everywhere and non-negative.

## 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 an integrable function g on S such that fn ≥ −g for all n, then

$\int _{S}\liminf _{n\to \infty }f_{n}\,d\mu \leq \liminf _{n\to \infty }\int _{S}f_{n}\,d\mu .$

#### Proof

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

### Pointwise convergence

If in the previous setting the sequence f1, f2, . . . converges pointwise to a function f μ-almost everywhere on S, then

$\int _{S}f\,d\mu \leq \liminf _{n\to \infty }\int _{S}f_{n}\,d\mu \,.$

#### Proof

Note that f has to agree with the limit inferior of the functions fn 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.

#### Proof

There exists a subsequence such that

$\lim _{k\to \infty }\int _{S}f_{n_{k}}\,d\mu =\liminf _{n\to \infty }\int _{S}f_{n}\,d\mu .$

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 with Varying Measures

In all of the above statements of Fatou's Lemma, the integration was carried out with respect to a single fixed measure μ. Suppose that μn is a sequence of measures on the measurable space (S,Σ) such that (see Convergence of measures)

$\mu _{n}(E)\to \mu (E),~\forall E\in {\mathcal {F}}.$

Then, with fn non-negative integrable functions and f being their pointwise limit inferior, we have

$\int _{S}f\,d\mu \leq \liminf _{n\to \infty }\int _{S}f_{n}\,d\mu _{n}.$

## 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 $(\Omega ,\,{\mathcal {F}},\,\mathbb {P} )$ ; the integrals turn into expectations. In addition, there is also a version for conditional expectations.

### Standard version

Let X1, X2, . . . be a sequence of non-negative random variables on a probability space $(\Omega ,{\mathcal {F}},\mathbb {P} )$  and let ${\mathcal {G}}\,\subset \,{\mathcal {F}}$  be a sub-σ-algebra. Then

$\mathbb {E} {\Bigl [}\liminf _{n\to \infty }X_{n}\,{\Big |}\,{\mathcal {G}}{\Bigr ]}\leq \liminf _{n\to \infty }\,\mathbb {E} [X_{n}|{\mathcal {G}}]$    almost surely.

Note: Conditional expectation for non-negative random variables is always well defined, finite expectation is not needed.

#### 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.

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

$Y_{k}=\inf _{n\geq k}X_{n}.$

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

$\mathbb {E} [Y_{k}|{\mathcal {G}}]\leq \mathbb {E} [X_{n}|{\mathcal {G}}]$    almost surely

by the monotonicity of conditional expectation, hence

$\mathbb {E} [Y_{k}|{\mathcal {G}}]\leq \inf _{n\geq k}\mathbb {E} [X_{n}|{\mathcal {G}}]$    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 Yk, the monotone convergence theorem for conditional expectations, the last inequality, and the definition of the limit inferior, it follows that almost surely

{\begin{aligned}\mathbb {E} {\Bigl [}\liminf _{n\to \infty }X_{n}\,{\Big |}\,{\mathcal {G}}{\Bigr ]}&=\mathbb {E} [X|{\mathcal {G}}]=\mathbb {E} {\Bigl [}\lim _{k\to \infty }Y_{k}\,{\Big |}\,{\mathcal {G}}{\Bigr ]}=\lim _{k\to \infty }\mathbb {E} [Y_{k}|{\mathcal {G}}]\\&\leq \lim _{k\to \infty }\inf _{n\geq k}\mathbb {E} [X_{n}|{\mathcal {G}}]=\liminf _{n\to \infty }\,\mathbb {E} [X_{n}|{\mathcal {G}}].\end{aligned}}

### Extension to uniformly integrable negative parts

Let X1, X2, . . . be a sequence of random variables on a probability space $(\Omega ,{\mathcal {F}},\mathbb {P} )$  and let ${\mathcal {G}}\,\subset \,{\mathcal {F}}$  be a sub-σ-algebra. If the negative parts

$X_{n}^{-}:=\max\{-X_{n},0\},\qquad n\in {\mathbb {N} },$

are uniformly integrable with respect to the conditional expectation, in the sense that, for ε > 0 there exists a c > 0 such that

$\mathbb {E} {\bigl [}X_{n}^{-}1_{\{X_{n}^{-}>c\}}\,|\,{\mathcal {G}}{\bigr ]}<\varepsilon ,\qquad {\text{for all }}n\in \mathbb {N} ,\,{\text{almost surely}}$ ,

then

$\mathbb {E} {\Bigl [}\liminf _{n\to \infty }X_{n}\,{\Big |}\,{\mathcal {G}}{\Bigr ]}\leq \liminf _{n\to \infty }\,\mathbb {E} [X_{n}|{\mathcal {G}}]$    almost surely.

Note: On the set where

$X:=\liminf _{n\to \infty }X_{n}$

satisfies

$\mathbb {E} [\max\{X,0\}\,|\,{\mathcal {G}}]=\infty ,$

the left-hand side of the inequality is considered to be plus infinity. The conditional expectation of the limit inferior might not be well defined on this set, because the conditional expectation of the negative part might also be plus infinity.

#### Proof

Let ε > 0. Due to uniform integrability with respect to the conditional expectation, there exists a c > 0 such that

$\mathbb {E} {\bigl [}X_{n}^{-}1_{\{X_{n}^{-}>c\}}\,|\,{\mathcal {G}}{\bigr ]}<\varepsilon \qquad {\text{for all }}n\in \mathbb {N} ,\,{\text{almost surely}}.$

Since

$X+c\leq \liminf _{n\to \infty }(X_{n}+c)^{+},$

where x+ := max{x,0} denotes the positive part of a real x, monotonicity of conditional expectation (or the above convention) and the standard version of Fatou's lemma for conditional expectations imply

$\mathbb {E} [X\,|\,{\mathcal {G}}]+c\leq \mathbb {E} {\Bigl [}\liminf _{n\to \infty }(X_{n}+c)^{+}\,{\Big |}\,{\mathcal {G}}{\Bigr ]}\leq \liminf _{n\to \infty }\mathbb {E} [(X_{n}+c)^{+}\,|\,{\mathcal {G}}]$    almost surely.

Since

$(X_{n}+c)^{+}=(X_{n}+c)+(X_{n}+c)^{-}\leq X_{n}+c+X_{n}^{-}1_{\{X_{n}^{-}>c\}},$

we have

$\mathbb {E} [(X_{n}+c)^{+}\,|\,{\mathcal {G}}]\leq \mathbb {E} [X_{n}\,|\,{\mathcal {G}}]+c+\varepsilon$    almost surely,

hence

$\mathbb {E} [X\,|\,{\mathcal {G}}]\leq \liminf _{n\to \infty }\mathbb {E} [X_{n}\,|\,{\mathcal {G}}]+\varepsilon$    almost surely.

This implies the assertion.