In the definition of Radiance, which according to PBRT is defined as

flux per unit solid angle $d\omega$ per unit projected area $dA^\perp$

$$L_i=\frac{d\Phi}{d\omega\ dA^\perp}$$

My original assumption was that the $dA^\perp$ term is used to weaken the irradiance contribution due to the incident angle (which I assume is just Lambert's cosine law). However, in the rendering equation, we've already explicitly apply Lambert's law with the $\cos\theta$ term:
$$\int_\Omega L_i(\omega') \, f_r(\omega, \omega') \, \cos\theta \, d\omega'$$

I assume we arn't applying Lambert's law twice, so what exactly is this $dA^\perp$ term?

In the definition of Radiance, which according to PBRT is defined as

flux per unit solid angle $d\omega$ per unit projected area $dA^\perp$

$$L_i=\frac{d\Phi}{d\omega\ dA^\perp}$$

My original assumption was that the $dA^\perp$ term is used to weaken the irradiance contribution due to the incident angle (which I assume is just Lambert's cosine law). However, in the rendering equation, we've already explicitly apply Lambert's law with the $\cos\theta$ term:
$$\int_\Omega L_i(\omega_{i}) \, f_r(\omega_{i}\rightarrow \omega_{o}) \, \cos\theta_{i} \, d\omega_{i}$$

I assume we arn't applying Lambert's law twice, so what exactly is this $dA^\perp$ term?