Approximating Measurable Functions with Simple Functions | Measure Theory

Three fundamental convergence theorems enable interchanging limits and integrals: (1) Monotone Convergence Theorem (Beppo Levi): For non-negative measurable functions increasing pointwise, lim ∫f_n = ∫lim f_n. (2) Fatou's Lemma: ∫lim inf f_n ≤ lim inf ∫f_n. (3) Dominated Convergence Theorem (Lebesgue): If f_n → f a.e. and |f_n| ≤ g with g integrable, then lim ∫f_n = ∫lim f_n. These replace the strong uniform convergence requirement of Riemann integrals with weaker conditions.

This section covers three fundamental convergence theorems essential for Lebesgue integration theory. The Monotone Convergence Theorem states that if a sequence of integrable functions is monotone almost everywhere and their integrals are uniformly bounded, then the limit function is integrable and the convergence holds in norm and almost everywhere. The Lebesgue Dominated Convergence Theorem provides a powerful tool for interchanging limits and integrals: if a sequence converges almost everywhere and is dominated by an integrable function, then the limit is integrable and the convergence holds in norm. Fatou's Lemma offers a weaker but still useful result for non-negative functions, ensuring that the integral of the limit inferior is less than or equal to the limit inferior of integrals. These theorems form the analytical foundation for modern integration theory and are extensively used in subsequent proofs throughout the course.

This lecture covers three fundamental convergence theorems in Lebesgue integration theory. The Monotone Convergence Theorem (MCT) states that for a sequence of non-negative measurable functions that are monotonically increasing, the integral of the limit equals the limit of the integrals. A weaker version allows the monotonicity condition to hold almost everywhere rather than everywhere. Fatou's Lemma provides an inequality (integral of lim inf ≤ lim inf of integrals) for any sequence of non-negative measurable functions, without requiring monotonicity. The Dominated Convergence Theorem (DCT) is the most general, requiring only that the sequence be dominated by an integrable function and converge pointwise, allowing interchange of limit and integral. The lecture demonstrates that the domination condition in DCT is essential through counterexamples, and shows that assumptions can be weakened to hold almost everywhere rather than everywhere.

This lecture demonstrates three fundamental convergence theorems in measure theory: (1) Lebesgue Dominated Convergence Theorem is applied to evaluate lim(n→∞)∫₀¹ n^(3/2)x/(1+n²x²)dx = 0 by showing the sequence is dominated by the integrable function g(x) = 1/√x; (2) Bounded Convergence Theorem is used to prove lim(n→∞)∫₀¹ n/(1+n²x²)dx = 0 by establishing uniform boundedness by 1/2; (3) Monotone Convergence Theorem is applied to compute ∫₀¹ x^(-1/3)dx = 3/2 by constructing an increasing sequence of non-negative bounded functions that converge to the unbounded function.

Lebesgue's Dominated Convergence Theorem states that if f_n converges pointwise to f and is dominated by an integrable g, then ∫f_n → ∫f. A counterexample f_n(x) = n·χ_[0,1/n](x) shows pointwise convergence alone is insufficient. The Monotone Convergence Theorem requires f_n ≤ f_{n+1} and uniformly bounded integrals. The proof constructs g_n(x) = inf_{k≥n} f_k(x), which converges to f(x). Since g_n ≤ f_n and ∫f_n converges, ∫g_n is bounded and converges. By the monotone convergence theorem, ∫g_n → ∫f. A corollary states that if f_n are non-negative integrable functions increasing to f and ∫f_n → C, then f is integrable and ∫f = C.

Egorov's theorem states that if a sequence of functions converges almost everywhere on a set of finite measure, then it converges uniformly on a subset of arbitrarily large measure. Applied to the difference quotient converging to 0, this allows us to find a set where convergence is uniform. Lusin's theorem states that a measurable function is continuous on a closed set of arbitrarily large measure. Applied to ∇f, this allows us to find a closed set Cⱼ where ∇f is continuous. The union of these sets C = ∪Cⱼ is closed and has measure arbitrarily close to full measure.

Applying Egorov's Theorem to the sequence of continuous functions that converge almost everywhere to f, we find for any ε > 0 an open set A with measure less than ε where the sequence converges uniformly on its complement. Taking E = complement of A, we have measure(E^c) < ε and f is continuous on E because the uniform limit of continuous functions is continuous. This completes the proof of Lusin's Theorem, showing that measurable functions are continuous on large sets.

Lusin's theorem states that for any measurable function f on a finite measure space, and for any ε > 0, there exists a continuous function g such that the set where f ≠ g has measure less than ε. This means measurable functions are nearly continuous. Combined with Egorov's theorem, these results form Littlewood's three principles: (1) measurable sets are nearly intervals, (2) measurable functions are nearly continuous, and (3) convergent sequences of measurable functions are nearly uniformly convergent. These principles provide intuitive characterizations of measure-theoretic concepts in terms of classical mathematical objects, showing that measure theory differs from classical analysis only on sets of small measure.

Littlewood's three principles state that every measurable set is nearly a finite union of intervals, every measurable function is nearly continuous, and every pointwise convergent sequence of measurable functions is nearly uniformly convergent. Egoroff's theorem proves the third principle by showing that for a sequence of measurable functions converging pointwise on a set of finite measure, there exists a closed subset where convergence is uniform and the complement has arbitrarily small measure. Lusin's theorem proves the second principle by showing that any measurable function can be approximated by a continuous function on a closed set with arbitrarily small complement measure.

This lecture covers fundamental theorems in real analysis: Egoroff's Theorem states that for a sequence of measurable functions converging pointwise almost everywhere on a finite measure space, the convergence is uniform except on a set of arbitrarily small measure; Lusin's Theorem establishes that measurable functions are nearly continuous, meaning for any ε > 0, there exists a closed set of measure at least μ(E) - ε where the function is continuous. These theorems are essential for understanding the relationship between pointwise convergence and uniform convergence, and for approximating measurable functions by continuous functions.

To prove density in lp, given x in lp and ε > 0, since the p-norm series converges, there exists M such that the tail sum from M to ∞ is less than ε/2. Then, using the density of Q^M in R^M, find a rational sequence q in the first M coordinates such that the p-norm of the difference in the first M coordinates is less than ε/2. The total p-norm difference is less than ε. This proves that A is dense in lp, and since A is countable, lp with 1 ≤ p < ∞ is separable.

For an open set ω ⊂ R^n equipped with Lebesgue measure, the set S of simple functions vanishing outside sets of finite measure is dense in L^p(ω) for 1 ≤ p < ∞. A simple function φ is defined as a function from ω to R that vanishes outside a set of finite measure. For 1 ≤ p < ∞, φ ∈ S if and only if φ ∈ L^p(ω). The density follows from approximating non-negative f ∈ L^p(ω) by increasing sequences of simple functions ϕ_n ≤ f, then using the Dominated Convergence Theorem to show ||ϕ_n - f||_p → 0. For general f, write f = f⁺ - f⁻ and approximate each part separately.

The space of simple functions is dense in L^p(X, μ) for 1 ≤ p < ∞. This means any function in L^p can be approximated arbitrarily closely by simple functions in the L^p norm. The proof follows similar reasoning as in L^1 case, using truncation and approximation techniques to construct approximating simple functions.

LP spaces are vector spaces of functions defined on RN with codomain in complex numbers, where a function f belongs to LP if the integral of |f(x)|^P with respect to Lebesgue measure is finite. The P-norm is defined as ||f||_P = (∫|f(x)|^P dx)^(1/P). To make this a proper norm, functions are identified up to equivalence classes where two functions are equivalent if they differ only on a set of Lebesgue measure zero. The space of continuous functions with compact support in RN is dense in LP for any P ≥ 1, meaning any P-integrable function can be approximated arbitrarily closely by continuous functions with compact support in the P-norm. This denseness result is proven using simple function approximation and Lebesgue's dominated convergence theorem, leveraging the convexity of the function λ → λ^P.

Lebesgue spaces L^p (for 1 ≤ p < ∞) are complete normed spaces (Banach spaces), meaning every Cauchy sequence converges to a limit function within the space. Additionally, simple functions are dense in L^p, meaning any function in L^p can be approximated arbitrarily closely by simple functions in the L^p norm. This density property allows extending integration theory from simple functions to general measurable functions through approximation arguments.