all regions of the source plane that produce lensed images of interest. For instance, if one were interested in quadruply imaged sources, the integral would be over all source positions that result in four image lenses. The quantity $\sigma$ is called the *lensing cross section, and of particular interest to us will be the cross sections $\sigma^{(N)}$ for producing $N$-image systems.*

In reality, one always has some flux limit $F_{min}$ which corresponds to a minimum source luminosity $L_{min}$. Fortunately, the above argument is easily generalized: let $dn_s(L,z_s)/dL$ be the number density of background sources with luminosity $L$. Then, the mean number of lensing events becomes $$\frac{dN_{lenses}}{dz_sdz_hd\Omega} = n_h\frac{d\chi}{dz_s}\frac{d\chi}{dz_h}
    \int d^2{\mathbf{y}} \int_{L_{min}/\mu({\mathbf{y}})}^\infty dL\ \frac{dn_s(L,z_s)}{dL}.$$ If the source luminosity function can be approximated by a power law $dn_s(L,z_s)/dL = AL^{-\alpha}$ (note both $A$ and $\alpha$ can depend on $z_s$), the above expression reduces to $$\frac{dN_{lenses}}{dz_sdz_hd\Omega} = n_b(>L_{min}) n_h 
    \frac{d\chi}{dz_s} \frac{d\chi}{dz_h} \sigma_B({\mathbf{p}},\alpha,z_h,z_s)
\label{eq:bcs}$$ where $n_b(>L_{min},z_s)$ is the number density of sources above the flux limit *in the absence of lensing, and $\sigma_B$ is given by $$\sigma_B ({\mathbf{p}},z_h,z_s,\alpha)= \int d^2{\mathbf{y}}\ \mu({\mathbf{y}})^{\alpha-1}$$ where $\mu({\mathbf{y}})$ is the total magnification of a source at position ${\mathbf{y}}$. Following @hutereretal05, we call $\sigma_B$ the *biased cross section. Indeed, since 