I have the following difference equation

$$
\tilde{u}\_k = \begin{cases}
u\_k & \text{if $\gamma\_k = 1$, no signal lost} \\
\tilde{u}\_{k-1} & \text{if $\gamma\_k = 0$, signal lost}
\end{cases}
$$

which can be rewritten to $\tilde{u}\_k = (1 - \gamma\_k) \tilde{u}\_{k-1} + \gamma\_k u\_k$. The chance that $\gamma\_k = 1$ is $p\_\gamma$ and the chance that $\gamma\_k = 0$ is $1 - p\_\gamma$ for all $k$. As a result we model $\gamma$ using a Markov chain.

Now I am a bit confused on what the random variable/element is in this case, the probability space $(\Omega,F,P)$ and measurable space $(E,\Sigma)$ is? 

Clearly $\gamma\_k$ is a discrete-time stochastic process which satisfies the Markov property. Further $P$ is clear to me, $P(\gamma = 1) = p\_\gamma$ and $P(\gamma = 0) = 1 - p\_\gamma$, but $\Omega = F = \{\text{signal lost},\text{no signal lost}\}$? And $\gamma : \Omega \rightarrow E$ with $E = \{0,1\}$? And what would $\Sigma$ be?