Yes (mainly). This follows because, if $0\le u\le1$ and $u_*:=1_{f<0}$, then $$uf-u_*f=(u-u_*)f\ge0,$$ whence $$\int u_*f\le\int uf.$$
The minimizer is not unique, though.
Iosif Pinelis
- 127.7k
- 8
- 107
- 229