Fluctuating backgrounds and measurements of faint sources. The model Gaussian source 𝒢 with a size of 10″ (FWHM) and a peak intensity of 1 corresponds to the flat background ℬ₁ = 1 (red lines). The same source 𝒢 is also added to the nonuniform (hill- and hollow-like) backgrounds ℬ_1.9, ℬ_1.5, ℬ_0.5, and ℬ_0.1 (blue and green lines). The fluctuating backgrounds were obtained from the flat background ℬ₁ by adding or subtracting the Gaussians with a size of 15″ (FWHM) and the peak values of 0.5 and 0.9. Extraction methods would not be able to recognize that the real backgrounds are hill- or hollow-like, hence they would instead subtract the flat backgrounds, based on the intensities just outside the apparent source footprints. Therefore, the source 𝒢 would be extracted with over- or underestimated FWHM sizes A, peak intensities F_P, and total fluxes F_T (the middle, left, and right columns of numbers, respectively).