Let $f:\mathbb{R}^n\to \mathbb{R}$ be a smooth and bounded function which tends to 0 at infinity. Define, for $t>0$, the distribution $$ \nu (t) = \int \limits _{f(x)\ge t} dx, $$ and (in the distributional sense) the positive measure $$ \mu = -\frac{d\nu }{dt}. $$ By a change of variables we see that $$ \langle \mu , \phi \rangle _{\mathcal{D}', \mathcal{D}}=\int \limits _{\mathbb{R}^n} \phi (f(x))\,dx \quad \text{for }\phi \in C_0^\infty (\mathbb{R}^n). $$ In the text I'm reading it is stated (without motivation or proof) that $$ \operatorname{supp}\operatorname{sing}_A \mu = \operatorname{supp}\operatorname{sing}_A \nu , $$ where $\operatorname{supp}\operatorname{sing}_A $ denotes the analytical singular support of a distribution (i.e. the smallest closed set outside of which the distribution is real analytic). In general we have $\operatorname{supp}\operatorname{sing}_A u' \subset \operatorname{supp}\operatorname{sing}_A u$ but why do we have equality in a case like this?

Let $f:\mathbb{R}^n\to \mathbb{R}$ be a smooth and bounded function which tends to 0 at infinity. Define, for $t>0$, the distribution $$ \nu (t) = \int \limits _{f(x)\ge t} dx, $$ and (in the distributional sense) the positive measure $$ \mu = -\frac{d\nu }{dt}. $$ By a change of variables we see that $$ \langle \mu , \phi \rangle _{\mathcal{D}', \mathcal{D}}=\int \limits _{\mathbb{R}^n} \phi (f(x))\,dx \quad \text{for }\phi \in C_0^\infty (\mathbb{R}_+). $$ In the text I'm reading it is stated (without motivation or proof) that $$ \operatorname{supp}\operatorname{sing}_A \mu = \operatorname{supp}\operatorname{sing}_A \nu , $$ where $\operatorname{supp}\operatorname{sing}_A $ denotes the analytical singular support of a distribution (i.e. the smallest closed set outside of which the distribution is real analytic). In general we have $\operatorname{supp}\operatorname{sing}_A u' \subset \operatorname{supp}\operatorname{sing}_A u$ but why do we have equality in a case like this?