probability - Expected value of the minimum of a non-negative random variable and a constant

X is a non-negative random variable. Define Y = MIN(X, c) where c is a constant. What is E[Y]?
I am modeling the constant as another random variable whose pdf is Dirac Delta function: $f_{c}(x) := \delta(x-c)$. The mean and variance of this "constant random variable"(!) comes out as c and 0, but does this approach have enough mathematical rigor?

Answer

Assume that $c > 0$ otherwise $\min(X,c) = c$ which is rather trivial. Then $Y := \min(X,c)$ is a nonnegative variable and we have

$$\begin{align*} E[Y] &= \int_0^\infty P( Y > t) dt \\ &= \int_0^\infty P( X > t, c > t) dt \\ &= \int_0^\infty P( X > t)1\{ c > t\} dt \\ &= \int_0^c P(X > t)dt \end{align*}$$