Why a Conditional Fatou Lemma?
Fatou's lemma compares the expectation of a pointwise lower limit with the lower limit of expectations. Its conditional version makes the same comparison while retaining information in a sigma-algebra \(\mathcal G\). The result is useful when a sequence of conditional expectations converges: it gives a lower bound on that limit in terms of the conditional expectation of the pointwise lower limit.
The comparison is one-sided. Even if the conditional expectations converge, they need not converge to the conditional expectation of the pointwise limit. The conditional Fatou lemma gives a robust lower bound without assuming an integrable dominating random variable.
Statement and Conventions
Work on a probability space \((\Omega,\mathcal F,\mathbb P)\), and let \(\mathcal G\subseteq\mathcal F\) be a sigma-algebra. For a nonnegative random variable \(X\), \(\mathbb E[X\mid\mathcal G]\) denotes a \(\mathcal G\)-measurable version of its conditional expectation. We allow \(X\) and its conditional expectation to take the value \(+\infty\); the conditional expectation is characterized by \(\int_A \mathbb E[X\mid\mathcal G]\,d\mathbb P=\int_A X\,d\mathbb P\) for every \(A\in\mathcal G\).
The relevant earlier result is conditional monotone convergence: if \(0\leq U_n\uparrow U\), then \(\mathbb E[U_n\mid\mathcal G]\uparrow\mathbb E[U\mid\mathcal G]\) almost surely. We also use order preservation: \(U\leq V\) almost surely implies \(\mathbb E[U\mid\mathcal G]\leq\mathbb E[V\mid\mathcal G]\) almost surely for nonnegative random variables. These facts are applied below to a countable sequence, so their almost-sure comparisons can be made simultaneously outside one null set.
Proof of the Conditional Fatou Lemma
For \(n\in\mathbb N\), define the infimum of the \(n\)-th tail by \(Y_n=\inf_{k\geq n}X_k\). The sequence \(Y_n\) is nonnegative and increases pointwise, and its limit is \(\liminf_{n\to\infty}X_n\). For every \(k\geq n\), \(Y_n\leq X_k\). Order preservation therefore gives \(\mathbb E[Y_n\mid\mathcal G]\leq\mathbb E[X_k\mid\mathcal G]\) almost surely.
There are only countably many pairs \((n,k)\) with \(k\geq n\), so all these comparisons hold together outside one null set. Taking the infimum over \(k\geq n\) yields, on that set, \(\mathbb E[Y_n\mid\mathcal G]\leq\inf_{k\geq n}\mathbb E[X_k\mid\mathcal G]\). Now apply conditional monotone convergence to \(Y_n\uparrow\liminf_n X_n\). The left side increases almost surely to \(\mathbb E[\liminf_n X_n\mid\mathcal G]\); the right-side tail infima increase to \(\liminf_n\mathbb E[X_n\mid\mathcal G]\). Passing to these limits proves the inequality.
\(\square\)
Worked Examples
Worked Example: Fatou's Inequality Can Be Strict
Let \(\Omega=(0,1)\) with Lebesgue probability measure, and take the trivial sigma-algebra \(\mathcal G=\{\varnothing,\Omega\}\). Set \(X_n(x)=n\,\mathbf 1_{(0,1/n)}(x)\). For each fixed \(x\in(0,1)\), once \(n>1/x\), the indicator is zero. Hence \(X_n(x)\) is eventually zero and \(\liminf_n X_n(x)=0\). Its conditional expectation given the trivial sigma-algebra is therefore zero.
On the other hand, \(\mathbb E[X_n\mid\mathcal G]=\mathbb E[X_n]=n(1/n)=1\) for every \(n\). Thus the conditional Fatou comparison is \(0\leq1\), strictly. The expectations remain positive because the values grow as the sets on which they are nonzero shrink; pointwise, every fixed outcome eventually leaves those sets.
Worked Example: A Strict Inequality on One Conditioning Event
Let \(\Omega=\{a,b,c,d\}\), with each point having probability \(1/4\), and let \(\mathcal G\) be generated by the partition \(\{\{a,b\},\{c,d\}\}\). For odd \(n\), set \(X_n(a)=0\), \(X_n(b)=2\); for even \(n\), set \(X_n(a)=2\), \(X_n(b)=0\). For all \(n\), set \(X_n(c)=2\) and \(X_n(d)=4\).
Conditional expectation on each cell of this finite partition is the average over its two points. On \(\{a,b\}\), that average is always \((0+2)/2=1\); on \(\{c,d\}\), it is always \((2+4)/2=3\). Thus \(\liminf_n\mathbb E[X_n\mid\mathcal G]\) equals \(1\) on the first cell and \(3\) on the second.
At both \(a\) and \(b\), the sequence takes the value zero infinitely often, so \(\liminf_n X_n=0\) at each. At \(c,d\), its values are constantly \(2,4\). Consequently, \(\mathbb E[\liminf_n X_n\mid\mathcal G]\) is \(0\) on \(\{a,b\}\) and \((2+4)/2=3\) on \(\{c,d\}\). The comparison is strict on the first cell and an equality on the second.
Lower Bounds When Conditional Expectations Converge
By the conditional Fatou lemma, the conditional expectation on the left is at most \(\liminf_n\mathbb E[X_n\mid\mathcal G]\) almost surely. On the event where the assumed convergence to \(Z\) holds, the lower limit of the conditional expectations is \(Z\). This proves the first claim. If \(X_n\to X\) almost surely, then \(\liminf_nX_n=X\) almost surely; conditional expectations of almost-surely equal nonnegative variables are almost surely equal, giving the second claim.
\(\square\)
Worked Example: A Limit of Conditional Expectations Bounds a Conditional Lower Limit
Use \(\Omega=\{a,b,c,d\}\), the uniform probability measure, and the sigma-algebra generated by \(\{a,b\}\) and \(\{c,d\}\). For odd \(n\), let \((X_n(a),X_n(b))=(0,4)\); for even \(n\), let \((X_n(a),X_n(b))=(2,2)\). Set \(X_n(c)=3\) and \(X_n(d)=5\) for every \(n\).
On \(\{a,b\}\), \(\mathbb E[X_n\mid\mathcal G]=(0+4)/2=2\) for odd \(n\), and \((2+2)/2=2\) for even \(n\). On \(\{c,d\}\), it is always \((3+5)/2=4\). The conditional expectations therefore converge to \(Z\), equal to \(2\) and \(4\) on the respective cells.
The pointwise lower limits are \(0\) at \(a\), \(2\) at \(b\), \(3\) at \(c\), and \(5\) at \(d\). Their conditional averages are \((0+2)/2=1\) on the first cell and \((3+5)/2=4\) on the second. Hence \(\mathbb E[\liminf_nX_n\mid\mathcal G]\) equals \(1\) and \(4\), respectively, and is bounded above by \(Z\), as the corollary asserts.
What the Inequality Does—and Does Not—Say
The conditional Fatou lemma is valuable because it requires only nonnegativity. It does not require the \(X_n\) to be integrable, and it does not require pointwise convergence. If the sequence does converge almost surely, the corollary supplies a lower bound using its limit. If only the conditional expectations converge, the same conclusion applies to the pointwise lower limit.
A common mistake is to replace the inequality by an equality. The first example shows that this is not valid, even for the trivial sigma-algebra; the second shows that strictness can occur on a particular conditioning event. Convergence of conditional expectations alone does not establish that their limit equals the conditional expectation of a pointwise limit. Additional hypotheses are needed for such a stronger conclusion.
Another detail is the role of null sets. Conditional expectations are defined only up to almost-sure equality. The theorem asserts that versions can be compared almost surely; it does not assign significance to their values on exceptional null sets. Countability matters in the proof: it lets us combine the comparisons for all tail indices while still discarding only one null set.
Check Your Understanding
Use the tail-infimum proof and the worked examples to answer these questions.
- How does the sequence \(Y_n=\inf_{k\geq n}X_k\) relate to \(\liminf_nX_n\)?
- Where in the proof is conditional monotone convergence used?
- In the finite-partition example with alternating values on \(\{a,b\}\), why is the conditional expectation of each \(X_n\) equal to \(1\) there?
- If \(\mathbb E[X_n\mid\mathcal G]\) converges almost surely to \(Z\), what does the corollary bound by \(Z\)?
- Why does conditional Fatou not, by itself, give equality when \(X_n\) converges almost surely?