Likelihood Function of Random Process

Given the following data: $$x(t) = A + \omega(t)$$ where $\omega(t)$ is an AWGN with zero mean, what would be likelihood function $p(x(t);A)$?

I know it could be proven to be: $$p(x;A) = C \exp\left(- \frac{\int (x(t) - A))^2 dt}{2\sigma^2}\right)$$

Yet I don't know the reason and the formal proof.

Note:

Both RHS and LHS depends on the whole random process.

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What is an AEGN? Something Something Gaussian Noise? –  Chris Taylor Oct 27 '11 at 7:42
Additive white Gaussian noise. –  Emre Oct 27 '11 at 9:27
E = white?   –  Did Oct 27 '11 at 9:35
Apparently, E is a typo since on some keyboards W and E are close. Could you please check if my correction fits the sense of the question. –  Ilya Oct 27 '11 at 10:48
Drazick: the LHS of the last equation in your post depends on $x(t)$ while the RHS depends on $(x(s))_{0\leqslant s\leqslant t}$. You might wish to explain. –  Did Oct 27 '11 at 10:54