Relation of distributions and measures Suppose we have the Dirac "function" and consider it as a distribution. Then
\begin{equation}
f(x)=\int f(x-t)\delta(t)dt
\end{equation}
for suitable functions on $\mathbb{R}^n$. Can we also do this with the dirac measure? and is there a relation between the two? This measure is defined as ${\displaystyle \delta _{x}(A)=\left\{{\begin{matrix}1,&{\textrm {if}}\ x\in A\\0,&{\textrm {if}}\ x\,\not \in \,A\end{matrix}}\right.}$ and one could expect that $\delta(t)dt$ and $\delta_{t}( \{ t \} )$ would be a similar thing. Is there a way to connect these things?
Notice that this is not the same as Dirac delta distribution and measure?
where the $x$ in my notation is in the dirac "function". They should equal the same number.
On https://en.wikipedia.org/wiki/Dirac_measure we find the expression
$\int _{X}f(y)\,\mathrm {d} \delta _{x}(y)=f(x)$
what I am looking to have is the $x$ inside the $f$ instead but am not sure what kind of framework that is the best.
 A: The way you write your question, it seems to me like you're more or less just caught up in notation. I think you'd be hardpressed to find an actual, mathematical difference between the Dirac $\delta$-distribution and the Dirac measure $\delta_0$.
Let me elaborate a bit. You define a distribution $\delta$ on $C_c^{\infty}(\mathbb{R}^d)$ by
$$\langle \delta,f\rangle=f(0)$$
This is your defining feature of the Delta distribution. If you are more comfortable with notation that's indicative of integration you could write this as
$$
\int f(x-t) \delta(t)\textrm{d}t=f(x),
$$
but this is really just notation. The mathematically precise formulation would be the one above:  $\delta$ is the functional that returns the value of $f$ at $0$.
Now, for any suitable measure $\mu$ on $\mathbb{R}^d$, we get the functional
$$
\langle \mu,f\rangle=\int f\textrm{d}\mu
$$
which, for $\delta_0$, the Dirac mass at $0$, gives you
$$
\langle \delta_0,f\rangle=\int f\textrm{d}\delta_0=f(0)
$$
so in the world of distributions, $\delta=\delta_0$. There is no difference.
A: You can write, equivalently
$$ \int f(x-t) \,\mathrm{d}\delta_0(t) = \int f(t) \,\mathrm{d}\delta_x(t) $$
and these expressions consider $\delta_x$ as a measure. Sometimes, a bit improperly, these measures are written as functions i.e. in the form
$$ \int f(x-t)\delta_0(t)\,\mathrm{d} t = \int f(x)\delta_x(t)\,\mathrm{d} t $$
but it is important to remember that $\delta_0$ and $ \delta_x$ are not functions.
