TITLE: Questions on atoms of a measure
QUESTION [18 upvotes]: In Kai Lai Chung's A course in
probability theory,

An atom of any probability measure $\mu$ on
  $(\mathbb{R}, \mathcal{B})$ is a
  singleton $\{x\}$ such that $\mu({x}) > 0$.

In Wikipedia:

Given a measurable space $(X,Σ)$ and a
  measure $\mu$ on that space, a set A
  in Σ is called an atom if $\mu (A) >0$
  and for any measurable subset B of A
  with $\mu(A) > \mu (B)$, one has
  $\mu(B) = 0$.

I was wondering if Chung's and Wikipedia's definitions agree with each other? If yes, does Chung's
definition mean that in the special probability measure space
$(\mathbb{R}, \mathcal{B}, \mu)$, there
is no  atom, as defined as in
Wikipedia, that is not a singleton?
Chung says in his book that the
number of atoms of any
$\sigma$-finite measure is
countable. 

I was wondering if suppose there are    uncountably many atoms, how
will it    contradicts
$\sigma$-finite measure?
Is this conclusion also true for
general measure space $(X,Σ, \mu)$
instead of just for probability
measure space $(\mathbb{R},
        \mathcal{B}, \mu)$?


Thanks and regards!

REPLY [2 votes]: Take a look at the exercise 22 in the page 32 of the Chung's book. In this exercise, he asks that be demonstrated the equivalence between these two definitions of atoms when the measure is defined on the euclidean Borel field.