> [!definition] > > Let $X$ be a [[n-Manifold|n-manifold]] and $\omega \in \Lambda^n(X)$ be a continuous [[Differential Form|n-form]]. Then there exists a unique [[Positive Linear Functional|positive linear functional]] $\phi_\omega \in C_c(X)^*$ such that for any [[Atlas|chart]] $(U, \varphi)$ with > $ > \omega(x) = f(x)dx_1 \wedge \cdots \wedge dx_n > $ > and $g \in C_c(U)$, > $ > \angles{\phi_\omega, g} = \int g \circ \varphi^{-1}(x)\abs{f(x)}dx > $ > where the integral is taken with respect to the [[Lebesgue Measure|Lebesgue measure]]. The functional $\phi_\omega$ is the **functional associated with** $\abs \omega$. By the [[Riesz Representation Theorem (Radon Measure)|Riesz Representation Theorem]], there exists a unique [[Radon Measure|Radon measure]] $\mu_{\abs{\omega}}$ on $X$ such that > $ > \angles{\phi_\omega, g} = \int g d\mu_{\abs{\omega}} > $ > for all $g \in C_c(X)$, known as the **measure associated with** $\abs{\omega}$. > > *Proof*. For each chart $(U, \varphi = (x_1, \cdots, x_n))$, this defines a positive linear functional on $C_c(U)$. It's sufficient to check compatibility: Let $(V, \psi = (y_1, \cdots, y_n))$ be another chart such that $\omega = h dy_1 \wedge \cdots \wedge dy_n$ and $g \in C_c(U \cap V)$, then from the change of variables formula, > $ > \int g \circ \varphi^{-1}(x)\abs{f(x)}dx = \int g \circ \psi^{-1}(y)\abs{f \circ \varphi \circ \psi^{-1}(y)} \cdot \abs{D(\varphi \circ \psi^{-1})(y)}dy > $ > Here, by a property of [[Alternating Tensor|alternating tensors]], > $ > dx_1 \wedge \cdots \wedge dx_n = \det (\varphi \circ \psi^{-1})dy_1 \wedge \cdots \wedge dy_n > $ > so > $ > fdx_1 \wedge \cdots \wedge dx_n = f \circ \varphi \circ \psi^{-1} \cdot \det (\varphi \circ \psi^{-1})dy_1 \wedge \cdots \wedge dy_n > $ > which allows writing > $ > \int g \circ \varphi^{-1}(x)\abs{f(x)}dx = \int g \circ \psi^{-1}(y)\abs{h(y)}dy > $ > Thus our definition is coordinate-free. By the [[Gluing Lemma for Linear Functionals]], this defines a unique linear functional on $C_c(X)$ with the desired properties. > [!theorem] > > Let $X$ be an $n$-manifold and $\omega \in \Lambda^n(X)$ be a continuous $n$-form. Then there exists a unique functional $\phi_\omega \in C_c(X)^*$ such that for any oriented chart $(U, \varphi)$ where > $ > \omega = f dx_1 \wedge \cdots \wedge dx_n > $ > and for any $g \in C_c(U)$, > $ > \angles{\phi_\omega, g} = \int g \circ \varphi^{-1}(x) \cdot f(x) dx > $ > where the integral is taken with respect to the Lebesgue measure. The functional $\phi_\omega$ is known as the **functional associated with** $\omega$, and the induced measure $\mu_\omega$ is the **measure associated with** $\omega$. > > *Proof*. Oriented = drop the absolute signs.