Here is a question about decomposition of measures in singular parts and in positive and negative parts.

$\newcommand{\RR}{\mathbb{R}}$ Let $\Omega_{1/2}$ be compact subsets of $\RR^d$ equipped signed measures $\alpha_{1/2}$. Let $\lambda$ denote the Lebesgue measure on $\Omega_{1/2}$. We decompose $$ \alpha_i = f_i + \eta^+_i-\eta^-_i $$ where $f_i$ is the Radon-Nikodym derivative of $\alpha_i$ with respect to $\lambda$, and $\eta^+_i-\eta^-_i$ is the Hahn-Jordan decomposition of of the respective singular parts. Put differently, we decompose both $\alpha_i$s into an $L^1$-function, a positive singular measure and a negative singular measure.

Further define the *outer sum* of $\alpha_1$ and $\alpha_2$ by
$$
\alpha_1\oplus\alpha_2 = \alpha_1\times \lambda + \lambda\times\alpha_2.
$$

Does it hold that $$ (\alpha_1\oplus\alpha_2)^+ = \Big((f_1 + \eta^+_1)\oplus(f_2 + \eta^+_2)\Big)^+, $$ in other words, is the positive part of $\alpha_1\oplus\alpha_2$ unaffected, if we omit the negative part of the singular parts?

Is suspect that the answer is affirmative, but I have trouble calculating the positive part (I tried using the characterization $\mu^+(A) = \sup_{E\subset A}\mu(A)$ but the choice of $E$ get quite messy…)