Defined on $[0,1] $ or $(0,1) $ About the support of uniform distribution I think it is not too dumb a question..  
I get a definition of uniform distribution from wikipedia. It says that uniform distribution is defined such that density is 1 at the interval $[0,1]$ and zero everywhere. Well, if we integrate them throughout all real line, Indeed we get the integration 1. However, if we consider another density which is 1 at the interval $(0,1)$ and zero everywhere. We still get the integration 1. It seems it is also a valid probability density.. My question is that is this new density also uniform distribution?   
Maybe more general, how about density with value 1 in $(0,1]$ or in $[0,1)$.
 A: The uniform distribution is a continuous one, thus the probability of any particular value is $0$.  It does not matter if the bounds are strict or not, it is still the same distribution.
$$P(X=x)=0, \forall x$$ so it does not matter if you include $P(X=0)$ or $P(X=1)$ as these are both $0$
A: Let $\mathsf{P}$ be a probability measure on $\mathbb{R}$ s.t. $\mathsf{P}(B)=m(B\cap[0,1])$ for a Borel set $B$, where $m$ is the Lebesgue measure on $\mathbb{R}$ ($\mathsf{P}$ corresponds to the uniform prob. distribution on $[0,1]$). Since $\mathsf{P}\ll m$, there exists the Radon-Nikodym derivative $f=d\mathsf{P}/dm$, which is unique up to $m$-null sets (in this case $f=1_{[0,1]}$ a.e.). It means that any $g$ s.t. $m(\{g\ne 1_{[0,1]}\})=0$ is a version of the density of $\mathsf{P}$ w.r.t. $m$ (for example, we may take $g=1_{[0,1]\setminus \mathbb{Q}}$).  
A: It's the same distribution either way, since its integral over any set is the same.
But since you mention "support", we should look at the definition of that term. A number $a$ is a member of the support of a distribution if that distribution assigns positive probability to every interval $(a-\varepsilon,a+\varepsilon),$ no matter how small $\varepsilon>0$ is.
That implies $0$ and $1$ are both members of the support of this distribution, so the support is $[0,1].$ The support is always a closed set.
