The situation for a Lambertian surface (emitting or scattering) is illustrated in Figures 1 and 2.  For conceptual clarity we will think in terms of photons rather than energy or luminous energy. The wedges in the circle each represent an equal angle dΩ, of an arbitrarily chosen size, and for a Lambertian surface, the number of photons per second emitted into each wedge is proportional to the area of the wedge.
The length of each wedge is the product of the diameter of the circle and cos(θ). The maximum rate of photon emission per unit solid angle is along the normal, and diminishes to zero for θ = 90°. In mathematical terms, the radiance along the normal is I photons/(s·m·sr) and the number of photons per second emitted into the vertical wedge is . The number of photons per second emitted into the wedge at angle θ is .
Figure 2 represents what an observer sees. The observer directly above the area element will be seeing the scene through an aperture of area dA and the area element dA will subtend a (solid) angle of dΩ, which is a portion of the observers total angular field-of-view of the scene. Since the wedge size d'Ω was chosen arbitrarily, for convenience we may assume without loss of generality that it coincides with the solid angle subtended by the aperture when "viewed" from the locus of the emitting area element dA. Thus the normal observer will then be recording the same  photons per second emission derived above and will measure a radiance of
:<math>
I_0=\frac{I\, d\Omega\, dA}{d\Omega_0\, dA_0}
·sr).
The observer at angle θ to the normal will be seeing the scene through the same aperture of area dA (still corresponding to a dΩ wedge) and from this oblique vantage the area element dA is foreshortened and will subtend a (solid) angle of dΩ cos(θ).  This observer will be recording  photons per second, and so will be measuring a radiance of
:<math>
I_0=\frac{I \cos(\theta)\, d\Omega\, dA}{d\Omega_0\, \cos(\theta)\, dA_0}
=\frac{I\, d\Omega\, dA}{d\Omega_0\, dA_0}
·sr),
which is the same as the normal observer.