Special case for Riesz Representation Theorem

Question

The usual Riesz Representation theorem can be found at page 49 in this book.

Here I modify it a little bit and providing the following version Riesz representation theorem, where I replaced $\mathbb R^m$ in original theorem by $\mathbb S^{m\times m}$ which appears below. ($\mathbb S^{m\times m}$ denotes $m\times m$ symmetric matrix)

Let $L$: $C_c(\mathbb R^n;\mathbb S^{m\times m})\to\mathbb R$ be a linear functional satisfying $$ \sup\{ L(f),\,f\in C_c(\mathbb R^n;\mathbb S^{m\times m}), |f|\leq 1,\text{ spt}(f)\subset K\}<\infty $$ for each compact set $K\subset\mathbb R^n$. Then there exists a Radon measure $\mu$ on $\mathbb R^n$ and a $\mu$-measurable function $\sigma$: $\mathbb R^n\to\mathbb S^{m\times m}$ such that

1. $|\sigma(x)|=1$ for $\mu$-a.e. $x$, and

2. $L(f)=\int_{\mathbb R^n}f:\sigma \,d\mu$

for all $f\in C_c(\mathbb R^n;\mathbb S^{m\times m})$. Here $f:\sigma$ is the frobenius product.

Is there anybody know whether my new Riesz theorem is correct or not? I think it is right but I can not find a reference.

Thank you!