Is "probability distribution function" a distribution? I can understand the definition of distribution as written in https://en.wikipedia.org/wiki/Distribution_(mathematics)  On the other hand there are three different terms in the definition of probability distribution function(PDF) :  https://en.wikipedia.org/wiki/Probability_distribution_function
My question: is PDF a distribution? If so can anyone help me to clarify how a PDF is a distribution? 

The only thing that relates them are that they are both restricted cases of measures (or at least the PDF can be interpreted as such). 
A distribution in probability theory is very like a measure with the restriction that the measure of the whole space is 1 (ie $\int dp = $int p(x) dx = 1$).
The other distribution is quite restricted since it's only allowed to act on smooth functions with compact support (ie $\int \varphi d\mu$ need only bee defined if $\varphi$ is smooth with compact support).
But since a probability distribution is that general and smooth functions with compact support is so well behaved you can always integrate a such (ie $\int \varphi dp$ is well defined as required of the second type of distribution). You could of course generalize the concept of PDF to allow for any measure (that is not necessarily representable as a function).
