# Proof of a concept over Distributions

I'm approaching the study of distributions, and together with many notes, I'm following L.Hormander book "linear partial differential operators".

In page $$5$$ he writes

In view of the identification of an absolutely continuous measure with its density function, which is customary in integration theory, this means in particular that a function $$f \in L^1_{loc}(\Omega)$$ is identified with the distribution

$$\phi \to \int \phi f \ \text{d}x$$

This distribution will also be denoted by $$f$$.

Well my question is: is there a proof for that? Why a function $$f$$ is / can be identified with that distribution?

The fact that this defines a distribution should be rather straightforward from the definition since $$\left|\int \phi(x)\,f(x)\,dx\right|\leq\sup_{x\in K}|\phi(x)|\int_K |f(x)|\,dx$$ where $$K$$ is the support of $$\phi$$. So that if $$\phi_n\rightarrow\phi$$ in $$\mathcal{D}$$ then $$\int\phi_n(x)\,f(x) dx\longrightarrow\int \phi(x) f(x)\,dx$$
You can identify the distribution with the function because of the following fact: if $$f\in L^1_{loc}(\Omega)$$ and $$\int \phi(x) f(x)\,dx=0$$ for every $$\phi\in\mathcal{D}$$ then $$f=0$$ a.e. This shows that the mapping $$L^1_{loc}(\Omega)\rightarrow\mathcal{D'}(\Omega)$$ which assigns $$f$$ to the distribution you've defined is one-to-one, and we can consider $$L^1_{loc}(\Omega)$$ as embedded in the space of distributions.