probability - Expected value of non-negative bounded random variable

I have the random variable $Y$ and it is such that $0\leq Y\leq 1$ I want to bound its expected value, given that I have a high probability bound:
$P(Y\geq x) \leq g(x)$, can I say that:

$$E[Y] = \int_0^1P(Y\geq x) dx \leq \int_0^1 g(x) dx\, ?$$
I tried showing it but I am not really strong in measure theory and Fubini theorem:
$$\begin{align}\int_0^1P(Y\geq x) dx &= \int_0^1 \left(\int_y^\infty f_Y(z)dz\right)dy \\ &= \int_0^1 \left(\int_0^z f_Y(z)dy\right)dz \\ &= \int_0^1 z\, f_Y(z)dz = E[Y] \end{align}$$
Is it correct?