Lp Spaces and Fubini’s Theorem

Why Function Spaces Are Needed

After the introduction of the Lebesgue integral, it is natural to ask: which functions “behave nicely”? Why do we need a class of functions at all, rather than just individual functions? The answer is that analysis problems rarely exist in “a single function.” We approximate functions by series, seek solutions to equations in a function class, and study the convergence of sequences of functions. For all this, we need a metric space of functions—a set with a notion of distance that allows us to talk about closeness, convergence, and completeness.

Lp spaces are the proper functional spaces for analysis, probability theory, signal processing, and quantum mechanics.

Lᵖ Spaces: Definition and Norm

Let (X, μ) be a measure space. For 1 ≤ p < ∞:

Lᵖ(X, μ) is the space (of equivalence classes) of measurable functions f: X → ℝ such that ∫_X |f(x)|ᵖ dμ(x) < ∞.

Norm: ‖f‖_p = (∫_X |f|ᵖ dμ)^{1/p}.

Technically, L^p consists of equivalence classes: f ~ g if f = g almost everywhere (μ-a.e.). This makes ‖f − g‖_p = 0 equivalent to f = g a.e., which is necessary for the norm to be non-degenerate.

Space L^∞(X, μ): functions with finite essential norm ‖f‖_∞ = esssup|f| = inf{M: |f| ≤ M a.e.}. This is the limit of ‖f‖_p as p → ∞.

Concrete cases:

• L²([0,1]): functions with finite “mean squared” norm. Square-integrable functions.
• l²: sequences (aₙ) with Σ|aₙ|² < ∞. Countable analogue of L².
• L¹([0,1]): absolutely integrable functions. Norm = ∫₀¹|f|dx.

Hölder and Minkowski Inequalities

Hölder’s inequality: If f ∈ Lᵖ, g ∈ Lq, where 1/p + 1/q = 1 (p and q are conjugate exponents), then fg ∈ L¹ and:

‖fg‖₁ = ∫|fg| dμ ≤ ‖f‖_p · ‖g‖_q.

This generalizes the Cauchy–Bunyakovsky inequality (case p = q = 2): ∫|fg| ≤ √(∫f²) · √(∫g²).

Application: For p = 2, q = 2: |E[XY]| ≤ √(E[X²]) · √(E[Y²])—an inequality for moments in probability theory.

Minkowski’s inequality: ‖f + g‖_p ≤ ‖f‖_p + ‖g‖_p. This is the triangle inequality for the Lᵖ norm—it is what guarantees that Lᵖ is a normed space.

Riesz–Fischer theorem (completeness): Lᵖ is a Banach space (a complete normed space). Any sequence converging in norm has a limit in Lᵖ. This property does not hold for the space of continuous functions C[a,b] with the Lᵖ norm.

L² Space: Hilbert Structure

L²(X, μ) stands out among all Lᵖ: for p = 2, the norm is induced by an inner product:

(f, g) = ∫_X f(x) g(x) dμ(x), ‖f‖₂ = √(f,f).

L²(X, μ) is a Hilbert space: a Banach space with an inner product. This is an infinite-dimensional analogue of Euclidean space ℝⁿ.

Orthonormal basis in L²[−π, π]: the functions {1/√(2π), cos(nx)/√π, sin(nx)/√π}_{n≥1} form an orthonormal basis. Expansion as a Fourier series is expansion in the Hilbert space basis L².

Parseval’s equality: ‖f‖₂² = |a₀|²/2 + Σ(|aₙ|² + |bₙ|²)—the “Pythagorean theorem” in infinite-dimensional space.

In quantum mechanics: the system’s state is a vector in L²(ℝ³) (wave function ψ), ‖ψ‖₂ = 1 (normalization). Observables are self-adjoint operators in L².

Fubini and Tonelli Theorems

For multiple Lebesgue integrals, the central result is the possibility of changing the order of integration.

Tonelli’s theorem (for non-negative functions): If f ≥ 0 is measurable on X × Y (with σ-finite measures μ, ν), then:

∫_{X×Y} f d(μ×ν) = ∫_X [∫_Y f(x,y) dν(y)] dμ(x) = ∫_Y [∫_X f(x,y) dμ(x)] dν(y),

and both repeated integrals are defined (possibly = +∞) without any conditions.

Fubini’s theorem (for integrable functions): If ∫_{X×Y} |f| d(μ×ν) < ∞, then:

1. For μ-almost all x, the function y ↦ f(x,y) is integrable with respect to ν.
2. The function x ↦ ∫_Y f(x,y) dν(y) is integrable with respect to μ.
3. ∫_{X×Y} f d(μ×ν) = ∫_X [∫_Y f(x,y) dν(y)] dμ(x).

Warning: without the integrability condition, Fubini can yield different answers for different orders! Classic example: f(x,y) = (x²−y²)/(x²+y²)² on [0,1]×[0,1]. ∫₀¹∫₀¹ f dx dy = π/4 ≠ −π/4 = ∫₀¹∫₀¹ f dy dx. The reason: ∫∫|f| = +∞, the Fubini condition is not satisfied.

Radon–Nikodym Theorem and Conditional Expectation

Radon–Nikodym theorem: If ν ≪ μ (ν is absolutely continuous with respect to μ: from μ(E) = 0 follows ν(E) = 0), then there exists a unique function f ∈ L¹(μ) (the Radon–Nikodym derivative) such that ν(E) = ∫_E f dμ for all measurable E.

Notation: f = dν/dμ.

Applications in statistics. If a probability measure P “has a density” p(x) with respect to Lebesgue measure λ, then P ≪ λ and p = dP/dλ is the usual probability density function.

Conditional mathematical expectation E[X|G], where G is a σ-subalgebra, is defined as the unique G-measurable function Z with ∫_A Z dP = ∫_A X dP for all A ∈ G. This is an application of the Radon–Nikodym theorem—one of the culminating points of probability theory.

Question for thought: Show that continuous functions C[0,1] are dense in L²[0,1]: for any f ∈ L²[0,1] and ε > 0, there exists a continuous g with ‖f − g‖₂ < ε. Why is this important for numerical analysis?