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Below is a question from an old real analysis exam that I need help in solving.

Let $\mu, \nu$, and $\lambda$ be $\sigma$-finite, nonnegative and nontrivial measures on the measure space $(X,\mathcal{M})$ such that for every measurable set $A\in \mathcal{M}$, $$\mu(A) = \nu(A) + \lambda(A).$$ Then I want help in showing the following:

(i) If $\nu$ and $\lambda$ are mutually singular, then $\exists$ $B\in \mathcal{M}$ such that $\nu(A) = \mu(A\cap B)$.
(ii) If $\nu \ll \lambda$, then $\displaystyle 0\leq \frac{d\nu}{d\mu} \lt 1$ a.e. $[\mu]$ on $X$.

PS: I have only just learnt signed measures and the Radon-Nikodym Theorem.


For (i) using @Robert's hints, since $\nu \perp \lambda$, there exists measurable sets $B$ and $B^c$ such that $\nu(B^c) = 0 =\lambda(B).$ Also, for any $A\in \mathcal{M}$, $\nu(A) = \nu(A\cap B)$ and $\lambda(A) = \lambda(A\cap B^c)$. Thus, for any set $A\in \mathcal{M}$, $$ \begin{align*} \mu(A\cap B) & = \nu(A\cap B) + \lambda(A\cap B)\\ & = \nu(A) + 0\\ & = \nu(A). \end{align*}$$

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1 Answer

up vote 2 down vote accepted

Hint for (i): take $B$ such that $\lambda(B)=0$ and $\nu(B^c) = 0$.

Hint for (2): if $C = \{x: \frac{d\nu}{d\mu}(x) \ge 1$, what can you say about $\nu(C)$, $\mu(C)$ and $\lambda(C)$?

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Thanks for your answer: I've added to my post regarding (i); is it alright? . Could you please elaborate on the hint for (ii)? Thanks –  Kuku Mar 16 '12 at 19:55
    
Further hint: $\nu(C) = \int_C \frac{d\nu}{d\mu}\ d\mu \ge \ldots$. –  Robert Israel Mar 17 '12 at 0:01
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