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In the setting of quantum mechanics, the Riesz Representation Theorem guarantees that quantum states (which are continuous linear functionals on a Hilbert space of quantum observables) can be uniquely associated with wavefunctions or density matrices.

Per the theorem, the measure associated with a linear functional is a regular Borel measure, which is both inner and outer regular, and this regularity is crucial for many analytical and topological properties.

The Riesz Representation Theorem states that every continuous linear functional on the space of continuous, compactly supported functions on a locally compact Hausdorff space can be represented as an integral with respect to a unique regular Borel measure.

The theorem shows that the Lebesgue measure is the unique measure that represents the integration functional on the space of continuous functions with compact support on the real line.

In the context of Hilbert spaces, the theorem implies that every continuous linear functional can be represented as an inner product with a unique element of the space, which leads to the famous Riesz representation for Hilbert spaces.

The Riesz Representation Theorem assures that the representing measure is unique, meaning that there is only one measure that correctly represents the linear functional.