Check if pmf/pdf is valid According to this and this, we can say that a function f is pmf/pdf iff f is non-negative and the sum/integral over... is 1.
But for example, for pmf, there is also a property called "countable additivity". Do we also have to check this property?
 A: My intuitive answer is no: Since you use a pmf, the outcome variable is discrete and as a result the outcome values represent disjoint sets by definition.
A: $\sigma$-additivity is not a property of probability mass function, but of probability per se. By the third axiom of probability:

Any countable sequence of disjoint sets (synonymous with mutually
  exclusive events) $E_1, E_2, \ldots$ satisfies  $$P\left(\bigcup_{i =
 1}^\infty E_i\right) = \sum_{i=1}^\infty P(E_i)$$

For discrete random variable, the probability that $X$ is equal any of the values in set $S$ is
$$
\Pr(X \in S) = \sum_{x \in S} f(x)
$$
and for continuous random variable
$$
\Pr(X \in S) = \int_{x \in S} f(x) \,dx
$$ 
what follows from properties of probabilities for mutually exclusive events. In fact, this is a part of definition of probability density function. Moreover, in continuous case it follows from the additivity of integrals:
$$
\int_a^c f(x)\,dx = \int_a^b f(x)\,dx + \int_b^c f(x)\,dx\
$$
In plain English, if random variable $X$ can take $x_1,x_2,x_3,\dots$ values, then probability of observing any of the $x_i$ values is the fraction of the area under the $f(x)$ curve that they occupy, where the total area is $\Pr(\Omega)=\Pr(\bigcup_{i=1}^\infty x_i)=1$. Fractions are non-negative and always smaller then the total area. "Masses" of mutually exclusive fractions are additive. So it is not about properties of probability mass functions, or probability density functions, but about basic axioms of probability.
