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In the path integral approach to gauge theory, observables are gauge invariant functions on the space $A$ of a $G$-connections on $E$, where $G$ denotes the structure group and $E$ the fiber bundle. Therefore, an observable $f$ is a function on the space $A/G$, of connections modulo gauge transformations.
As a result, vacuum expectation values are no longer defined as integrals with Lebesgue measure $A$, but instead with a Lebesgue measure on $A/G$. We obtain this measure by pushing forward the Lebesgue measure on $A$ by the map $A \to A/G$ that sends each connection to its gauge equivalence class, and then A$ denotes a gauge equivalence class of connections in the integral.
The simplest example of an observable in gauge theory are Wilson loops. (Source: Baez, Munian; Gauge Fields, Knots and Gravity, page 342)