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      Defining the value of a distribution at a point

      Let $\omega\in D'(\mathbb R^n)$ be a distribution and $p\in \mathbb R^n$. If there is an open set $U\subset \mathbb R^n$ containing $p$ such that $\omega|_U$ is given by a continuous function $f\in C(U)$, then for every $\phi\in C^\infty_c(\mathbb R^n)$ with $\int_{\mathbb R^n}\phi(x)d x=1$ we can define a Dirac sequence $\{\phi^p_j\}_{j\in \mathbb N}\subset D(\mathbb R^n)$ by $\phi^p_j(x):=j^n\phi(j(x-p))$ which fulfills $$ \omega(\phi^p_j)\to f(p)\quad \text{ as }j\to \infty. $$ This shows that we can recover the value $\omega(p)\equiv f(p)$ of the distribution $\omega$ at the point $p$ via a limit of such Dirac sequences.

      Now, suppose that for some $\omega\in D'(\mathbb R^n)$ and $p\in \mathbb R^n$ we just know that $ \lim_{j\to \infty}\omega(\phi^p_j) $ exists for every $\phi\in C^\infty_c(\mathbb R^n)$ with $\int_{\mathbb R^n}\phi(x)d x=1$ and is independent of $\phi$. In view of the above it then seems reasonable to define $\omega(p):=\lim_{j\to \infty}\omega(\phi^p_j)$ and to say that $\omega$ has a well-defined value at the point $p$.

      Q: Is this definition useful in any sense? I have the feeling that it might be fundamentally flawed. In that case, I'd find it interesting to know what's the greatest generality in which one can make sense of "the value of a distribution at a point".

      Additional thoughts after 1st edit: Some "consistency checks" for the definition would in my opinion be the following:

      1. If the value of $\omega$ exists at every point in some open set $U\subset \mathbb R^n$ and the function $f$ defined on $U$ by these values is continuous, then $\omega|_U$ is given by $f$.

      2. If the value of $\omega$ exists at Lebesgue-almost every point in some open set $U\subset \mathbb R^n$ and the values define a function $f\in L^1_{\mathrm{loc}}(U)$, then $\omega|_U$ is given by $f$.

      I believe that at least property 1 should be true and I'll check it once I find the time.

      2nd edit: My question is related to this MO question which corresponds to the case $f\equiv 0$.

      Answer

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