Non-atomic measure

Remarks

Lebesgue measure $\lambda$ on $\R$ is non-atomic. If $\lambda(A) > 0$, the function $t \mapsto \lambda(A \cap (-\infty, t])$ is continuous and increases from $0$ to $\lambda(A)$. So there is $t$ with $B = A \cap (-\infty, t]$ such that $\lambda(B) > 0$ and $\lambda(A \setminus B) > 0$. The Dirac measure $\delta_x$ is not non-atomic, since $\set{x}$ is an atom: its only measurable subsets are $\varnothing$ and $\set{x}$. Some authors define an atom by the condition that every measurable $B \subseteq A$ has $\mu(B) = 0$ or $\mu(B) = \mu(A)$. For sets $A$ of finite measure the two conditions agree.