self learning – MAP and MMSE Estimations

Let the observation $x$ has the following density:
begin{equation}
p(x|theta)=
begin{cases}
frac{1}{theta},& text{if } 0 < x leq theta\
0, & text{otherwise}
end{cases}
end{equation}

and the r.v $theta$ has:
begin{equation}
p(theta)=
begin{cases}
theta exp(-theta),& text{if } 0 leq theta\
0, & text{otherwise}
end{cases}
end{equation}

I try to find MAP and MMSE estimation of the parameter $theta$.

MAP:

$hat{theta} = argmax_{theta}$ $p(theta|x) = argmax_{theta}$ $p(x|theta)p(theta) = argmax_{theta}$ $exp(-theta)$ which gives $hat{theta} = 0$. But it seems $p(x|theta)$ is not well defined at $theta = 0$. Am I missing something here?

MMSE:

$hat{theta} = E(theta | x) = int_{0}^{infty} theta exp(-theta) ,dtheta = 1$. I want to verify if it’s correct.