Measure Theory 11: Lebesgue's Dominated Convergence Theorem Proof

The completeness of Lp spaces (Banach space property) is proven by constructing a limit function from a Cauchy sequence. For 1 ≤ p < ∞, given a Cauchy sequence (f_n), we decompose the measure space into sets of finite measure and construct a limit function f by summing differences. Fatou's lemma and the dominated convergence theorem are used to show that f_n converges to f in Lp norm. For p = ∞, uniform convergence on sets of finite measure is used. This completeness property makes Lp spaces suitable for analysis.

This section provides the complete proof that L^p(E) is complete for 1 ≤ p < ∞. The proof uses the criterion that every absolutely summable series ∑f_k converges to an element in L^p(E). Define g_n(x) = ∑_{k=1}^n |f_k(x)|. By triangle inequality, ||g_n||_p ≤ ∑||f_k||_p converges to some M. By Fatou's lemma, ∑|f_k(x)| < ∞ almost everywhere, so define f(x) = ∑f_k(x) when convergent. Since |∑f_k - f| ≤ g(x) and g ∈ L^p(E), dominated convergence gives ∑f_k → f in L^p(E). This establishes L^p(E) as a Banach space.

To prove Lp is a Banach space (complete normed space), we must show every Cauchy sequence converges to a limit in Lp. Let (f_n) be a Cauchy sequence in Lp. For each ε > 0, there exists N such that for all n, m ≥ N, ||f_n - f_m||_p < ε. By Minkowski's inequality, this implies (f_n) is Cauchy in measure. There exists a subsequence (f_{n_k}) that converges almost everywhere to some function f. One can show f ∈ Lp and f_n → f in Lp norm. The proof involves showing that the limit function f satisfies ∫|f|^p dμ < ∞ and that the convergence holds in the Lp norm. Thus, Lp is complete and hence a Banach space.

To prove Lp is complete, we show x ∈ Lp and (x^m) converges to x. From the Cauchy condition, for m ≥ N, ∑|x_i^m - x_i^n|^p < ε^p for all n ≥ N. Taking limit as n → ∞, ∑|x_i^m - x_i|^p ≤ ε^p, so x^m - x ∈ Lp. Since x^m ∈ Lp and Lp is closed under addition, x = x^m + (x - x^m) ∈ Lp. For convergence, d_p(x^m, x) = (∑|x_i^m - x_i|^p)^(1/p) ≤ ε for m ≥ N. Thus every Cauchy sequence in Lp converges within Lp, making it a complete metric space (Banach space).

The space lp with the lp norm is a Banach space. The proof follows the same two-step approach as for l∞: first, construct a candidate limit by taking coordinate-wise limits of the Cauchy sequence; second, prove that this candidate belongs to lp and that the original sequence converges to it. The key step uses the boundedness of Cauchy sequences and the fact that the coordinate-wise limits satisfy the lp condition.

Uniform integrability (UI) is defined for a collection C of random variables by sup_{X ∈ C} E[|X|1_{|X| > K}] → 0 as K → ∞. Equivalently, C is UI iff: (1) sup E[|X|] < ∞, and (2) for any ε > 0, there exists δ > 0 such that for any A with P(A) < δ, sup E[|X|1_A] < ε. Single integrable variables are always UI, and dominated collections (|X| ≤ Y for all X ∈ C with Y integrable) are UI. Vitali's Convergence Theorem states that for X_n ∈ L¹ converging to X in probability, the following are equivalent: (1) {X_n} is UI, (2) X ∈ L¹ and X_n → X in L¹, and (3) X ∈ L¹ and E[X_n] → E[X]. This generalizes the Dominated Convergence Theorem to convergence in probability.

Vitali's theorem characterizes when convergence in probability implies L1 convergence: a sequence in L1 converging in probability to X has L1 convergence if and only if it is uniformly integrable. The proof involves showing that uniform integrability ensures the limit exists in L1 (using Fatou's lemma and splitting expectations), and conversely, L1 convergence implies uniform integrability through truncation arguments and Vitali's theorem. Additionally, conditional expectations E[X|F_α] form a uniformly integrable family when X is in L1, as they are bounded in L1 by E[|X||F_α] and satisfy uniform integrability through monotonicity arguments.

Uniform integrability is a condition that generalizes the domination condition in the Dominated Convergence Theorem. A family of functions is uniformly integrable if it is bounded in L^1 and for every ε > 0, there exists δ > 0 such that ∫_E |f| dλ < ε for all f in the family and all measurable sets E with λ(E) < δ. The Vitali Convergence Theorem states that if f_n converges pointwise to f and is uniformly integrable, then ∫f dλ = lim ∫f_n dλ. This theorem is more general than the Dominated Convergence Theorem. The theory of integration can be extended to functions with values in Banach spaces (Bochner integration) and to locally compact groups using Haar measures. The convergence theorems are essential for studying L^p spaces and stochastic processes.

This section develops the theory of uniform integrability, essential for martingale convergence. A family {X_i} is uniformly integrable if sup_i E[|X_i|I(|X_i| > K)] → 0 as K → ∞, meaning large values are uniformly negligible. Examples include families dominated by an integrable random variable. Vitali's Convergence Theorem characterizes L¹ convergence: X_k → X_∞ in L¹ if and only if X_k → X_∞ in probability and {X_k} is uniformly integrable. This generalizes the dominated convergence theorem and provides a complete characterization of convergence in the mean.

Uniform integrability is a property of a family of measurable functions where for every ε > 0, there exists a δ > 0 such that for any measurable subset A with measure less than δ, the integral of |f| over A is less than ε, and this δ depends only on ε, not on the specific function. Vitali's Convergence Theorem states that if E is a set of finite measure, and {f_n} is a sequence of measurable functions that is uniformly integrable over E and converges pointwise almost everywhere to f, then f is integrable over E and the limit of the integrals equals the integral of the limit: lim_{n→∞} ∫_E f_n = ∫_E f. This theorem provides conditions under which the interchange of integration and limiting processes is valid.

This section introduces two fundamental theorems for integrating functions over product measure spaces. Tonelli's theorem applies to unsigned measurable functions and states that the integral over the product space equals both iterated integrals, with the additional guarantee that the resulting functions are measurable. Fubini's theorem extends this to complex-valued L¹ functions, requiring absolute integrability but allowing the same interchange of integration order. These theorems form the theoretical foundation for multidimensional integration and enable practical computation through repeated integration rather than direct evaluation over product spaces.

Os teoremas de Tonelli e Fubini estabelecem que, para espaços de medida σ-finitos e funções mensuráveis, a integral dupla em relação à medida produto pode ser calculada por integrais iteradas: para funções não negativas, o teorema de Tonelli garante que ∫∫f d(μ×ν) = ∫[∫f dμ]dν = ∫[∫f dν]dμ, enquanto para funções integráveis (não necessariamente não negativas), o teorema de Fubini estende esse resultado, permitindo a troca da ordem de integração sob condições de σ-finitude.

Fubini's Theorem states that integrals over product spaces can be computed as iterated integrals. Two versions exist: a simple one for bounded measurable functions where all integrals exist, and a general one for functions measurable with respect to the completed product sigma-algebra. In the general case, internal integrals exist only almost everywhere. Pathological cases exist where iterated integrals exist and are equal, yet the function is not integrable on the product space. Tonelli's Theorem states that if F is measurable and one iterated integral exists, then F is integrable on the product space. The proof uses truncation: level sets of non-negative functions are bounded and measurable. By Fubini's Theorem applied to these bounded functions, iterated integrals are equal. The integrals of level sets are uniformly bounded, and by Levi's Theorem (monotone convergence), the original function is integrable. Product measures for finite numbers of spaces are constructed by induction: multiply two measures, then multiply the result with the next measure. This order independence follows from Fubini's Theorem. For countable products of probability measures, the measure on the product space of infinite sequences is defined on cylinder sets (finite products of measurable sets times full spaces). The measure of a cylinder set is the product of measures of its bases. This construction requires probability measures (total measure 1) to ensure countable additivity. The subgraph of a non-negative measurable function F is {(x, t) : 0 ≤ t ≤ F(x)}. For bounded F, the Lebesgue integral equals the measure of its subgraph in X × R. An alternative formula is ∫₀^∞ m({x : F(x) ≥ t}) dt, where m is the measure on X. For unbounded functions, the integral is defined by limits of bounded truncations.

Fubini's Theorem is a fundamental result in measure theory concerning integration on product spaces. For two measure spaces (X, M1, μ) and (Y, M2, ν), if f is a measurable function on X × Y and f is integrable (belongs to L1), then: (1) For almost every x in X, the section fx is integrable with respect to ν; (2) For almost every y in Y, the section fy is integrable with respect to μ; (3) The iterated integrals are well-defined and belong to L1; (4) The double integral equals the iterated integrals: ∫∫f d(μ×ν) = ∫(∫f(x,y)dν(y))dμ(x) = ∫(∫f(x,y)dμ(x))dν(y). Tonelli's Theorem extends this to non-negative measurable functions without requiring integrability. The proof of Fubini's Theorem uses Tonelli's Theorem by decomposing f into positive and negative parts f = f⁺ - f⁻, then applying Tonelli's Theorem to each part to establish measurability and integrability of sections and the equality of integrals.

This section develops the theory of integrating over product spaces. The product measure μ×ν on X×Y satisfies (μ×ν)(A×B) = μ(A)ν(B). The Fubini-Tonelli theorem states that if f is integrable over the product space, then iterated integrals equal the double integral regardless of integration order. For sigma-finite measure spaces, Tonelli's theorem extends this: if either iterated integral of |f| is finite, then f is integrable and the equality holds. These theorems enable powerful techniques for evaluating multidimensional integrals by reducing them to one-dimensional problems.