Are there theorems or results to show that if for every $\varphi\in \mathcal{C}_0^\infty(\mathbb{R})$ we have, $\int_{\mathbb{R}} \varphi^k(x)\mu(dx) \leq C$
Then $\mu(dx) = f(x)dx$ and $f\in \mathcal{C}^{\tilde{k}}(\mathbb{R})$ ?? where $\tilde{k}$ and $k$ might be related somehow.
I mean, is it for example true that if, $\int_{\mathbb{R}} \varphi'(x)f(x)dx \leq C$ for all $\varphi\in \mathcal{C}_0^\infty(\mathbb{R})$ then $f\in \mathcal{C}(\mathbb{R})$ ??
Here $\mathcal{C}_0^\infty(\mathbb{R})$ means infinitely many times diff. with compact support.
Thanks a lot for your help!! :)