We consider the classical additive measurement-error model X=Y+Z, where the latent random variable Y has unknown distribution FY and the error Z
Nearby in the stack
has a known distribution. We develop direct estimators for three functionals of
FY
: (i)
FY(x)
at continuity points; (ii) interval probabilities
FY(y)−FY(x)
when
x<y
are continuity points; and (iii) the size of a jump at a prespecified discontinuity. We derive non-asymptotic bias and variance bounds, and establish asymptotic unbiasedness and consistency. Unlike previous work, we do not require
FY
to admit a density, have a mixture representation, or satisfy global Sobolev smoothness assumptions. The framework accommodates arbitrary latent distributions, including those with both discrete and continuous components, and distributions with multiple jumps. These results rely on a link between Fourier inversion theorems and the algebraic structure of a class of estimators proposed in Mynbaev, Martins-Filho and Henderson (2022). A simulation study evaluates feasible tuning procedures and, where available, compares the finite-sample performance of the proposed estimators with existing methods.