Strict Inequality In Fatou's Lemma

Strict Inequality in Fatou's Lemma: When Expectations Fall Short

Fatou's Lemma is a cornerstone result in measure theory and probability, providing a powerful tool for analyzing sequences of integrable functions. It states that the limit inferior of the expectation of a sequence of non-negative measurable functions is less than or equal to the expectation of the limit inferior of the sequence. However, the crucial question arises: when does strict inequality hold? Understanding the conditions under which this strict inequality occurs, i.e., when E[liminf f_n] < liminf E[f_n], is vital for applying Fatou's Lemma effectively and avoiding misinterpretations. This article delves into the intricacies of strict inequality in Fatou's Lemma, providing a comprehensive exploration of the underlying mechanisms and offering illustrative examples.

Introduction to Fatou's Lemma

Let's begin with a formal statement of Fatou's Lemma:

Fatou's Lemma: Let (X, Σ, μ) be a measure space, and let {f_n} be a sequence of non-negative measurable functions on X. Then

∫liminf f_n dμ ≤ liminf ∫f_n dμ

In probability theory, if we consider the expectation operator E instead of the integral ∫, the lemma becomes:

E[liminf f_n] ≤ liminf E[f_n]

This inequality holds true regardless of the properties of the sequence {f_n} beyond non-negativity and measurability. The power of Fatou's Lemma lies in its generality; it doesn't require convergence of the sequence {f_n} in any sense. However, this generality comes at a cost: the inequality might be strict, meaning that a gap might exist between the two sides.

Understanding Strict Inequality: The Role of Mass "Escape"

The core reason for strict inequality in Fatou's Lemma is the potential for "mass escape". Consider the sequence of functions. Imagine the mass represented by the integral or expectation of each function f_n. If, as n increases, significant portions of this mass "escape" to infinity or otherwise "disappear" from the limit, then the inequality becomes strict. The limit inferior of the expectations captures the remaining mass, while the expectation of the limit inferior captures only the mass that remains "concentrated" in the limit.

Illustrative Examples: Unveiling the Conditions for Strict Inequality

Let's examine some illustrative examples to clarify the conditions that lead to strict inequality.

Example 1: Shifting Mass

Consider a sequence of functions defined on the real line with Lebesgue measure. Let f_n(x) = 1_{[n, n+1]}(x), where 1_A(x) is the indicator function of set A, equal to 1 if x ∈ A and 0 otherwise. This means f_n(x) is 1 on the interval [n, n+1] and 0 elsewhere.

• Expectation: E[f_n] = ∫f_n(x)dx = 1 for all n. Therefore, liminf E[f_n] = 1.

• Limit Inferior: liminf f_n(x) = 0 for all x. Therefore, E[liminf f_n] = ∫0 dx = 0.

In this example, E[liminf f_n] = 0 < 1 = liminf E[f_n]. The mass of 1, initially concentrated on the interval [n, n+1], "moves" infinitely far to the right as n → ∞. The limit inferior of the functions is everywhere zero, reflecting the "escape" of the mass.

Example 2: Oscillating Functions

Consider the sequence f_n(x) = sin²(nx) on the interval [0, 2π] with Lebesgue measure.

• Expectation: E[f_n] = (1/2π) ∫₀²π sin²(nx) dx = 1/2 for all n. Thus, liminf E[f_n] = 1/2.

• Limit Inferior: The sequence f_n(x) oscillates rapidly as n increases. The limit inferior is not easily computed, but intuitively, as n tends to infinity, the oscillations become increasingly rapid, and the average value remains around 1/2. However, the function does not converge pointwise. In this example, analyzing the limit inferior directly is complex, but through more advanced techniques, it can be shown that E[liminf f_n] < 1/2. The strict inequality arises due to the rapid oscillations preventing convergence to a constant.

Example 3: Sequences with Unbounded Variance

Consider sequences where the variance of f_n becomes unbounded as n increases. This high variance indicates that there's increasing uncertainty in the value of f_n, and a significant portion of the mass may spread out to regions that won't contribute to the limit inferior.

When Equality Holds: The Role of Convergence

Fatou's Lemma becomes an equality under certain convergence conditions. If the sequence {f_n} converges pointwise almost everywhere to a function f, then the inequality becomes an equality. More formally:

Fatou's Lemma (Equality Case): If f_n → f almost everywhere and f_n ≥ 0 for all n, then

∫f dμ = lim ∫f_n dμ

This implies that if the sequence converges almost everywhere, there is no "mass escape", and the inequality becomes an equality. The Monotone Convergence Theorem and the Dominated Convergence Theorem offer more specific conditions ensuring convergence and thus equality in Fatou's Lemma.

Practical Implications and Applications

Understanding the conditions for strict inequality in Fatou's Lemma is crucial in various fields, including:

• Probability Theory: Analyzing convergence of random variables, especially in the context of stochastic processes.

• Statistical Mechanics: Studying the limiting behavior of systems with a large number of particles.

• Optimization Theory: Working with sequences of functions in optimization problems. Strict inequality can indicate suboptimal solutions.

• Ergodic Theory: Analyzing long-term averages of dynamical systems.

In each of these fields, it is vital to understand when mass escape occurs and how it affects the inequality.

Advanced Considerations: Beyond Pointwise Convergence

While pointwise almost everywhere convergence guarantees equality, weaker forms of convergence (like convergence in measure or convergence in probability) do not always guarantee equality. The behavior of the sequence {f_n} needs careful scrutiny in such cases. The intricacies of these weaker convergence modes and their relationship with strict inequality require advanced measure-theoretic tools.

Frequently Asked Questions (FAQ)

Q: Can we have strict inequality even if the sequence converges in L1?

A: Yes. Convergence in L¹ (i.e., lim E[|f_n - f|] = 0) implies that the expectation converges, but it does not guarantee pointwise convergence almost everywhere, which is necessary for equality in Fatou's Lemma.

Q: Are there any practical techniques to determine whether strict inequality holds without direct computation of the limit inferior?

A: Often, analyzing the variance of the sequence {f_n} can offer insights. Unbounded variance suggests the possibility of mass escape, hinting at strict inequality. Alternatively, exploring the behavior of the sequence under different types of convergence (pointwise, L¹ etc.) can provide clues.

Q: How does Fatou's Lemma relate to other convergence theorems, such as the Monotone Convergence Theorem and the Dominated Convergence Theorem?

A: The Monotone Convergence Theorem and Dominated Convergence Theorem provide stronger convergence results that imply equality in Fatou's Lemma under specific conditions (monotonicity and domination, respectively). Fatou's Lemma is more general but only provides an inequality.

Q: What if the functions are not non-negative?

A: If the functions are not non-negative, Fatou's Lemma cannot be directly applied. However, we can still utilize the properties of the positive and negative parts of the function. For any function f, we can decompose it into its positive and negative parts, f⁺ = max(f, 0) and f⁻ = max(-f, 0). Then f = f⁺ - f⁻. We can apply Fatou's Lemma separately to the sequences {f_n⁺} and {f_n⁻}, and then combine the results.

Conclusion: Mastering the Nuances of Fatou's Lemma

Fatou's Lemma is a fundamental tool, but its practical application demands a deep understanding of the conditions that lead to strict inequality. The concept of "mass escape" offers a powerful intuitive understanding of why the inequality can be strict. By recognizing the role of convergence, particularly pointwise almost everywhere convergence, and by carefully examining the behavior of sequences with unbounded variance, one can effectively utilize Fatou's Lemma and avoid misinterpretations arising from overlooking the possibility of strict inequality. Understanding these nuances is crucial for advanced work in measure theory, probability, and related fields. While the lemma provides a powerful lower bound, appreciating the conditions under which this bound is strict allows for a more refined and accurate analysis of the limiting behavior of sequences of functions.