Assume that $\mu$ a quasi invariant measure on a homogenous space $M$. Pick $k\in M$. The Dirac distribution $\delta_k$ at $k$ associated with $\mu$ is defined as the distribution such that $\langle \delta_k ,\phi \rangle\equiv \phi (k)$ for any text function $\phi$ on $M$. Equivalently, it is introduced as a "Dirac delta function" $\delta (k,\cdot )$ such that $\int_{M}d\mu (q)\delta (k,q)\phi (q)\equiv \phi (k)$.
My question is that: (1) what is the definition of $\delta (\cdot ,\cdot )$ itself? Could you please introudce me a reference for it?
(2) What is the difference between "Dirac distribution" and "Dirac delta function"?
Thank you in advance.