Another suggestion but please double check.
Since y = g(x). I.e.
y = x , x < L
y = x-L , x => L
Then
E(y) = E(g(x)) = \int {g(x) f(x)} dx = \int_{-\infty}^{L} {x f(x)} dx + \int_{L}^{\infty} {(x-L)f(x)} dx
where I am saying m(x) is equivalent to E(x).
I am a bit rusty so...