Here is question I tried to answer for some time - it seems to be straightforward, but I have trouble figuring it out.


Let $\Omega$ be a compact domain in $\mathbb{R}^n$. For any signed Borel measure $\mu$ on $\Omega$ let $\mu\_+$ denote its positive part (obtained by Hahn-Jordan decomposition). My question is:



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> Is taking the positive part a continuous operation, i.e. does $\mu^n\to \mu$ in the space signed Borel measures imply $\mu^n\_+\to \mu\_+$ in the same space?
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Of course, the answer would be positive if
$$\|w^n\_+-w\_+\|\_{\mathfrak{M}}\leq \|w\_n-w\|\_{\mathfrak{M}}$$
where $\|\cdot\|\_{\mathfrak{M}}$ denotes the variation norm, but I could not see this.



Is taking the positive part of a measure a continuous operation?
If $\mu =f\cdot \lambda$ for a positive measure $\lambda$ (i.e., $\mu(A)=\int\_A fd\lambda$), isn't then $\mu\_+= f\_+ \cdot\lambda$ (where $f\_+$ is the positive part $\max\{f,0\}$ of $f$) and $\|\mu\|=\int|f|d\lambda$? Then $\|\mu\_+-\nu\_+\| \le \|\mu-\nu\|$ just follows from Radon-Nikodym (applied to $\lambda=|\mu|+|\nu|$) and $|f\_+-g\_+|\le |f-g|$.


