Intuition for Integrating "Against a Test Function" in Distribution Theory

Question

I know that we can consider (locally integrable) functions and measures to be distributions via the relationships $$ \langle T_{f},\varphi\rangle=\int_{\mathbb{R}} f(x)\varphi(x) \, d\mu(x) $$ and $$ \langle T_{\mu},\varphi\rangle=\int_{\mathbb{R}} \varphi(x) \, d\mu(x) $$ and the fact that $$ \langle T_{f},\varphi \rangle = \langle T_{g} , \varphi\rangle \implies f=g \, a.e. $$ but I'm not completely sure how by integrating against a test function that we can recover the pointwise values of $f$ or $\mu$ (at least a.e.). My understanding is that the value of the distribution $T_{f}$ depend on the test function that it is being evaluated at and the values of the distribution determine the pointwise values of $f$ almost everywhere since evaluating the function at a point would involve a integrating against the limit of a sequence of test functions whose limit is not a test function (i.e. the Dirac distribution). My intuition tells me that the test function just serves as some sort of "smooth approximation to an indicator function" and then you can dilate the test function in the same manner you might change the support of an indicator function. Am I on the right track or am I missing something?

Answer 1

In some sense, this is one of those things that, yes, you can view functions as distributions but you don't necessarily gain anything from it per se. If anything, you lose the operation of point evaluation at a minimum, as you've noted already. Interpreting a function as a distribution is going from viewing that function by how it behaves pointwise to viewing that function solely by how it behaves when integrated against a test function (and don't underestimate how big a mental shift this is).

One example where interpreting a function as a distribution might be useful is taking distributional derivatives. Rememeber, under this interpretation, we don't care about how the function behaves pointwise (and by extension continuity, differentiability, etc.), only how it behaves when paired with a test function. For example if $T$ is a distribution, we take the distributional derivative of $T$ such that via a simple integration by parts, we arrive at $$ \langle T', \varphi\rangle = -\langle T ,\varphi'\rangle $$ From which you should get the basic idea, seeing how we made no assumptions about (the pointwise behavior) of $T$ itself, only about how it behaves in the context of $\varphi$. Basically the utility of distributions lies in it's ability to extend the operations available to a given space of functions to the operations defined on $C^{\infty}_{c}$. That being said, there is a significant tradeoff since you lose any information about the function itself, since the behavior of a distribution is entirely dependent on being integrated against a test function. While I think your intuition about smooth functions and indicator functions has some merit, I'm not sure it captures the essence of the idea, which is that the smooth function exists so that operations not available to the distribution can then be "passed on" to the smooth function when evaluated as an inner product.