Find the Lebesgue decomposition of of Lebesgue measure with respect to counting measure

Question

Suppose $m$ is a Lebesgue measure on $R$ and $\lambda$ is counting measure on $R$ both on the Lebesgue $\sigma$-algebra. Find the Lebesgue decomposition of $m$ with respect to $\lambda$.

Is this even possible? I've been trying to find this decomposition for at least 2 days now and I fear I have gotten tunnel vision. If somebody could help me out I'd greatly appreciate it.. Wouldn't we need the counting measure to be absolutely continuous wrt the Lebesgue measure for this to be possible? Because it isn't. If we take a set of one element, all it $E$, then $m(E)=0$ but $\lambda(E)=1$..

Am I just not understanding this or something? Thanks for the help

Answer 1

You want to write $m$ as the sum of two measures $m_1$ and $m_2$ with the property that

1. $m_1 << \lambda$
2. $m_2 \perp \lambda$

Note $\lambda(E) = 0 \implies E = \emptyset \implies m(E) = 0$ so that $m << \lambda$. Thus you might as well take $$m_1 = m,\qquad m_2 = 0.$$