# Is this the correct interpretation of specific intensity?

I've always been taught to define specific intensity as the following quantity: if I have a detector here next to me with an area element $$dA$$ and I detect a power element $$dP$$ from a source of solid angle element $$d\Omega$$ in the frequency band $$d\nu$$ then the specific intensity, by definition, is $$I_\nu = dP/(d\nu \cos{\theta}dA d\Omega)$$ where the factor $$\cos{\theta}$$ compensates for the orientation of $$dA$$ w.r.t the source. Or, using my paint skills:

Now I've been reading Lena's book on observational astrophysics and his definition is a little different. Instead of defining the specific intensity as the light incident onto $$dA$$ emerging from solid angle $$d\Omega$$, Lena defines specific intensity as the light passing through $$dA$$ into the cone $$d\Omega$$.

I can't quite wrap my head around why these two are equivalent. Why would they be?

I came up with one explanation of my own: suppose a pencil of light rays pass through an area element $$dA$$ into a cone of solid angle $$d\Omega$$ so that the pencil is incident upon a second element $$dA'$$. Then the solid angle of $$dA$$ w.r.t. $$dA'$$ is again $$d\Omega' = d\Omega$$ (I do not know enough of infitnitisimal geometric elements to know whether this assertion requires more background), such that the light inside the pencil emergent from $$dA$$ equals the light in the pencil incident upon $$dA'$$. That is, $$I_\nu dA d\Omega = I_\nu dA' d\Omega$$.

Is this the correct explanation?