Sample space of a random experiment can be either discrete or continuous.
Probability mass function is defined as a function from powerset of discrete sample space to $[0,1]$
The probability density function over a continuous sample space is defined as a function f(x) with the following properties
1) Integral overall sample space is 1.
2) non-negative over all elements of sample space.
The probability distribution function is a function which is equal to probability mass function over discrete sample space and is equal to a probability density function over continuous random spaces.
Concerning to probability distribution function, is the above definition valid and complete?
I think this is a very good question, because it appears to me that the term "probability distribution" is used rather inconsistently. Here are a few ways I've seen it used: 1) The distribution of a random variable $X$ is the probability measure on $\mathbb R$ induced by $X$ (in other words, $\mu_X(A) = P(X \in A)$); 2) Sometimes the term "distribution of a random variable $X$" refers to the cumulative distribution function (CDF) of $X$ (perhaps because the previous definition is too hard to explain); 3) Sometimes the term "probability distribution" is used interchangeably with the term "probability measure"; 4) Sometimes the term "distribution" refers to either a PMF or a PDF of a random variable.
I think this inconsistent use of the term "probability distribution" has caused some real confusion. I'm going to stick my neck out and say that 1) is the "true" definition of a probability distribution, but I am always eager to hear the opinions of people who know a lot about probability.