6  Kernels

Published

December 8, 2025

6.1 Definition

A kernel from (E, \mathcal E) to (F, \mathcal F) is function K: E \times \mathcal F \to \bar{\mathbb R_+} satisfying:

  1. for every B \in \mathcal F, the map x \mapsto K(x, B) is (E, \mathcal E) \to (F, \mathcal F) measurable.
  2. for every x \in E, the map B \mapsto K(x, B) is a valid measure on (F, \mathcal F).

6.1.1 Types

  1. Probability kernel: for all x \in E, K(x, F) = 1
  2. Finite kernel: for all x \in E, K(x, F) < \infty
  3. \sigma-finite kernel: for all x \in E, B \mapsto K(x, B) is \sigma-finite.
  4. \Sigma-finite kernel: K = \sum K_n for a sequence of finite kernels K_n.
  5. Bounded kernel: \sup_{x \in E} K(x, F) < \infty
  6. \sigma-bounded kernel: there exists a partition (F_n) of F such that for each n, \sup_{x \in E} K(x, F_n) < \infty.
  7. \Sigma-bounded kernel: K = \sum K_n for a sequence of bounded kernels K_n.
  8. Markov kernel: K: E \times \mathcal E\to \mathbb R such that for every x \in E, K(x, E) = 1.
  9. Sub-Markov kernel: K: E \times \mathcal E\to \mathbb R such that for every x \in E, K(x, E) \leq 1.

Note that for \sigma-finite kernels, the partition may change for each x, while for \sigma-bounded, the partition has to be uniform i.e. hold for all x.

Probability kernels \subset bounded kernels \subset \sigma-bounded kernels \subset \Sigma-bounded kernels.

No similar hierarchy (unlike in the measures case) exists for the _-finite kernels.

6.2 Operations

6.2.1 Composition

Let K: E, \mathcal F \to \mathbb R and L: F, \mathcal G \to \mathbb R then the product/composition kernel $KL:E, G R $ is a kernel.

KL(x, B)= \int K(x, dy) L(y, B)

6.2.2 Conditional integral

Let f\in \mathcal F_+ be a positive measurable function. Then Kf is a positive function in \mathcal E_+ Kf = \int K(x, dy) f(y)

6.2.3 Marginal measure

If \mu is a measure on \mathcal E, K: E, \mathcal F \to \mathbb R is a kernel, then for B \in \mathcal F \mu K(B) = \int \mu(dx) K(x, B)

is a measure on (F, \mathcal F)

The corresponding Lebesgue integral is:

\mu K[f] = \int \mu(dx) \int K(x, dy) f(y) = \mu[ Kf ]

This encapsulates the law of total expectation (\mu is marginal distribution of X, K is conditional distribution of Y|X, the marginal of Y is \mu K, and the conditional expectation is Kf).

6.2.4 Functions on product spaces

Let K be a \Sigma-finite kernel from (E, \mathcal E) to (F, \mathcal F). Then for every positive function f \in \mathcal E \otimes \mathcal F measurable with the product sigma algebra, Tf is a positive function in \mathcal E_+.

Tf(x) = \int_{y \in F} K(x, dy) f(x, y)

6.2.5 Kernel density

Let \nu be a \sigma-finite measure on (F, \mathcal F) and k \in (\mathcal E \otimes \mathcal F)_+ be a positive function measurable wrt the product sigma algebra. Then K is a valid kernel:

K(x, dy) = \nu(dy) k(x,y)

K(x, A) = \int \nu (dy)k(x, y)

6.2.6 Product measure

Let K: E, \mathcal F \to \mathbb R be \Sigma-finite and \mu be a measure on (E, \mathcal E). Then, \pi is a measure on the product space (E\times F, \mathcal E \otimes \mathcal F) defined by:

\pi f = \int \mu(dx) \int K(x, dy) f(x, y), \qquad f \in (\mathcal E \otimes \mathcal F)_+

If \mu is \sigma-finite and K is \sigma-bounded then \pi is \sigma-finite and is the unique measure on the product space satisfying: \pi(A \times B) = \int_A \mu(dx) K(x, B)