Projection (measure theory)

In measure theory, a projection, specifically a measurable projection, refers to a function that maps a measurable space formed by a product of sets to one of its constituent sets, while preserving measurability. This is a fundamental concept when dealing with product measures and the integrability of functions defined on product spaces.

More formally, let (X, A) and (Y, B) be measurable spaces. The projection map πX : X × Y → X is defined by πX(x, y) = x. Similarly, the projection map πY : X × Y → Y is defined by πY(x, y) = y.

For πX to be a measurable projection, we require that πX⁻¹(A) ∈ A ⊗ B for all A ∈ A, where A ⊗ B denotes the product σ-algebra of A and B. In other words, the pre-image of any measurable set in X under the projection πX must be a measurable set in the product space X × Y with respect to the product σ-algebra. This condition ensures that πX is a measurable function. The same holds true for πY.

Measurable projections are crucial in several aspects of measure theory, including:

• Product Measures: When defining product measures on product spaces, the measurability of projections is essential for ensuring that the defined measure is well-behaved and consistent with the underlying σ-algebras.

• Fubini's Theorem and Tonelli's Theorem: These fundamental theorems, which allow for iterated integration, rely heavily on the measurability of projections. These theorems specify conditions under which one can interchange the order of integration in a double integral, and the measurability of projections guarantees that the iterated integrals are well-defined.

• Marginal Distributions: In probability theory, which utilizes measure theory as its foundation, projections are used to define marginal distributions. If (X, Y) is a random vector with a joint distribution, the marginal distribution of X is obtained by "integrating out" Y. This operation is based on the projection of the joint probability space onto the space of X.

• Sections of Measurable Sets: The measurability of projections plays a role in understanding the measurability of sections of measurable sets in product spaces. A section of a set E ⊆ X × Y at a point x ∈ X is defined as Ex = {y ∈ Y : (x, y) ∈ E}. The projection property helps establish relationships between the measurability of E and the measurability of its sections.

In summary, the concept of a measurable projection in measure theory is a fundamental tool for analyzing measurable spaces formed by products of sets, especially in the context of product measures, integration, and probability theory. It guarantees that the "shadow" of a measurable set under the mapping remains measurable, facilitating various key results and constructions.