proving that a function is the density of an absolutely continuous distribution How does one go about proving that a function is the density of an absolutely continuous distribution. That is, what are the steps I need to go through to satisfy this definition?
For example, the standard cauchy distribution has $f(x):=\frac{1}{\pi}\frac{1}{1+x^2}$, $x\in\mathbb R$. What steps are required in this case?
Is it merely checking that it integrates to 1 over its potential x values?
 A: The original question asks [taking some liberties] for the conditions under which one can prove that an absolutely continuous function is in fact the probability density function (PDF) of a probability distribution.  While the definition of a probability distribution is useful, it is certainly not the whole answer.  We also need to look at the relationship between PDFs and distributions.
A probability distribution is a function from a set of events to the unit interval, where as $f$ in this case is from $\mathbb{R} \rightarrow \mathbb{R}$.  Points in $\mathbb{R}$ are not our events!  Measurable sets in $\mathbb{R}$ are our events.  So we attempt to define a distribution by defining the function
$$
P(A) = \int_A f d\mu = \int_A f(x)dx,
$$
the last bit of notation being included to make some people more comfortable.  Since $f$ is absolutely continuous (way more than measurable), $P$ is defined.  The first two axioms from Tim's post, follow immediately if we assume:
$$
\int_{\mathbb{R}} f d\mu = 1, \\
f(x) \geq 0, \forall x \in \mathbb{R}.
$$
Note that the second assumption there is stronger than we really need, but if $f$ is absolutely continuous, this is equivalent to the weaker statement $\int_A f d\mu \geq 0, \forall A$.  The last axiom falls out of the countable additivity of the Lebesgue integral, namely:
$$
P(\cup_i A_i) = \int_{\cup_{A_i}}fd\mu = \sum_i \int_{A_i}fd\mu = \sum_i P(A_i).
$$
This is one of those cases where, not so coincidentally, the structure of two mathematical objects (probability distributions and the Lebesgue integral) are in correspondence.  These are import moments in math, and they can really provide a lot of insight.
A: 
1.5 Definition. A function $\mathbb{P}$ that assigns a real number $\mathbb{P}(A)$ to each event $A$ is a probability distribution or a
  probability measure if it satisfies the following three axioms:  
Axiom 1: $\mathbb{P}(A) \geq 0$ for every $A$
  Axiom 2: $\mathbb{P}(\Omega) = 1$
  Axiom 3: If $A_1,A_2,...$ are disjoint then
  $\mathbb{P}(\bigcup_{i=1}^\infty A_i) = \sum_{i=1}^\infty\mathbb{P}(A_i)$

Source: Wasserman, L. (2004). All of Statistics. Springer.
