What is the meaning of Common Support here I am reading a notes in statistical inference, and I am constantly being confused about the term 'common support', i hardly find any definition of this,here is an example, 
'Suppose S is a space of all probability distributions with common support'
and the picture below is what I am reading, and it says ' here pdf of a family $\Bbb P$ of distributions must be of the commmon support of $\Bbb P $ '$(\Bbb P := \{P_\theta,\theta \in \Omega \})$
 A: The support of a Borel measure on $\mathbb R^n$, say, is defined as the set of points $x$ such that, for every $r\gt0$, $\mu(B(x,r))\gt0$. In your context, one considers a family $\{P_\theta;\theta\in\Theta\}$ and one asks that the support of $P_\theta$ does not depend on $\theta$.
A: In general, it means the set of all points where at least one density function is non-zero.  For example, if you have two uniform random variables, where one ranges from 0 to 1, and one ranges from 1 to 2, then the common support is 0 to 2.  (Sometimes "common support" means the topological closure of this set, but this probably doesn't matter for you.)
The "common support" assumption is important here because you can truncate an exponential family of random variables, to get a new exponential family of random variables.  For example, if you take the normal distribution (as a family with two parameters — the mean and variance) and force it to be between -3 and 5, what you get is no longer normally distributed, but it still forms an exponential family.  The common support is then [-3, 5].
Some exponential families are only defined on subsets of the reals, rather than the whole real line.  For example the family
$$
e^{-cx^3}
$$
(where $c$ is the parameter) is only defined for $x \geq 0$.  The common support in this instance is $[0, \infty)$.
