Research proof index

Given $a<b$ and $c<d$, there is a smooth compactly supported function $\Psi:\mathbb R\to\mathbb R$ with $\mathbf 1_{[b,c]}\le\Psi\le\mathbf 1_{(a,d)}$ and support exactly $(a,d)$. In the usual ordered case $a<b\le c<d$, this is a smooth cutoff equal to $1$ on $[b,c]$, strictly supported in $(a,d)$, and taking values between $0$ and $1$.

There exists a smooth nonnegative function $\nu:\mathbb R\to\mathbb R$ supported in $[1/2,2]$ and normalized to have multiplicative mass one: $\int_0^\infty \nu(x)\,\frac{dx}{x}=1.$