Riesz's representation of a k-current

Question

Reading "Introduction to GMT" by Simon, at page 136 he says that thanks to Riesz's representation theorem we can view k-currents ar Radon measures, or to be more precise he says that given a current $T$ on some open set $U$ , there's a Radon measure $\mu_T$ and a measurable function $\widetilde{T}$ such that for evey k-form $\omega$ on $U$:

$$ T(\omega)= \int_U \langle \omega (x) , \widetilde{T} (x) \rangle d \mu_T (x)$$

Now, the question is: what are the components of $\widetilde{T}$? Is there some "explicit" representation of this? We said in class that $\widetilde{T}$ has components $T_\alpha$ such that

$$ T_\alpha (f)= T (f dx_\alpha)$$

but I may have misunderstood, or anyway I didn't understand. Let's say I have:

$$ \omega (x) = f(x) dx_1 \wedge dx_2 + g(x) dx_2 \wedge dx_3 .$$

What is an explicit example of 2-current? Maybe in the form as above, seen as a vector valued Radon easure with some components.

EDIT: I forgot to say that $ T$ must have finite mass, otherwise it's false in general

Answer 1

Let $U$ be a open set of $\mathbb{R}^{3}$. For $\omega \in D_{2}(U)$ define 2-current $T(\omega)=\int_{U}d\omega$. Then the components of $\tilde{T}$ are

• $T_{12}(f)=T(fdx^{1} \wedge dx^{2})=\int_{U}\partial_{x^{3}}f dx^{3} \wedge dx^{1} \wedge dx^{2}=\int_{U}\partial_{x^{3}}fdx^{1}\wedge dx^{2} \wedge dx^{3}$.
• $T_{13}(f)=T(fdx^{1} \wedge dx^{3})=\int_{U}\partial_{x^{2}}f dx^{2} \wedge dx^{1} \wedge dx^{3}=-\int_{U}\partial_{x^{2}}fdx^{1}\wedge dx^{2} \wedge dx^{3}.$
• $T_{23}(f)=T(fdx^{1} \wedge dx^{2})=\int_{U}\partial_{x^{1}}f dx^{1} \wedge dx^{2} \wedge dx^{3}$.

For the specified $\omega$, $$T(\omega)=\int_{U}(\partial_{x^{3}}f +\partial_{x^{1}}g)dx^{1}\wedge dx^{2} \wedge dx^{3}.$$