# Definition:Trace Sigma-Algebra

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*This page is about trace $\sigma$-algebras. For other uses, see Definition:Trace.*

## Definition

Let $X$ be a set, and let $\Sigma$ be a $\sigma$-algebra on $X$.

Let $E \subseteq X$ be a subset of $X$.

Then the **trace $\sigma$-algebra (of $E$ in $\Sigma$)**, $\Sigma_E$, is defined as:

- $\Sigma_E := \left\{{E \cap S: S \in \Sigma}\right\}$

It is a $\sigma$-algebra on $E$, as proved on Trace Sigma-Algebra is Sigma-Algebra.

## Also known as

The **trace $\sigma$-algebra** may also be called the **trace sigma-algebra**, the **induced $\sigma$-algebra (on $E$)** or the **induced sigma-algebra (on $E$)**.

It is common to write $E \cap \Sigma$ for $\Sigma_E$, but this can cause confusion; hence it is discouraged on ProofWiki.

## Sources

- 2005: René L. Schilling:
*Measures, Integrals and Martingales*... (previous) ... (next): $3.3 \ \text{(vi)}$