Great question. The answer is that a likelihood function is just a function of random variables, so that the probability space that encapsulates all the probabilistic information of interest about those random variables (including the $\sigma$-fields you mention) is often still all that is required for a function of those random variables.
For example let $X_{1},X_{2}$ be random variables on the same underlying probability space $(\Omega,\mathscr{F})$ where $\mathscr{F}$ is a $\sigma$-field of subsets of $\Omega$. Then by definition $X_{i}:\Omega\longrightarrow\mathbb{R}$ for $i=1,2$ is a function that satisfies $X_{i}^{-1}(B)=\{\omega\in\Omega:X_{i}(\omega)\in B\}\in\mathscr{F}$ for all $B\in\mathscr{R}$ where $\mathscr{R}$ is the $\sigma$-field of subsets of $\mathbb{R}$. This is referred to as $X_{i}$ being $\mathscr{F}/\mathscr{R}$ measurable, "measurable" since $P$ (the probability measure) assigns a probability to the "event" $X_{i}\in B$ as $Pr[X_{i}\in B]=P[X_{i}^{-1}(B)]$. Thus a random variable is a "measurable mapping" between the measure spaces $(\Omega,\mathscr{F})$ and $(\mathbb{R},\mathscr{R})$.
Consider the (fixed) function $f:\mathbb{R}\times\mathbb{R}\longrightarrow\mathbb{R}$, $f(y,z)\mapsto y+z+37$ and use $X_{1}$ and $X_{2}$ as random arguments: $f(X_{1}(\omega_{1}),X_{2}(\omega_{2}))\mapsto X_{1}(\omega_{1})+X_{1}(\omega_{2})+37$. The question you ask of likelihood functions can be asked of any function. Can we assign probabilities to the event $[f(X_{1}(\omega_{1}),X_{2}(\omega_{2}))]^{-1}(B)=\{(\omega_{1},\omega_{2})\in \Omega\times\Omega:(X_{1}(\omega_{1})+X_{1}(\omega_{2})+37)\in B\}$ for all $B\in\mathscr{R}$ using the original space $(\Omega,\mathscr{F})$? The answer is yes if $\{(\omega_{1},\omega_{2})\in \Omega\times\Omega:(X_{1}(\omega_{1})+X_{2}(\omega_{2})+37)\in B\}\in\mathscr{F}$ since $P$ can assign probabilities to any $F\in\mathscr{F}$. If $f$ is continuous (here it is) then $[f(X_{1}(\omega_{1}),X_{2}(\omega_{2}))]^{-1}\in\mathscr{F}$ indeed holds. Thus $Pr[f(X_{1},X_{2})\in B]=P\left\{[f(X_{1},X_{2})]^{-1}(B)\right\}$
Thus viewing $f$ not as the mapping $f:\mathbb{R}\times \mathbb{R}\longrightarrow\mathbb{R}$, $f(y,z)\mapsto y+z+37$ but instead as $f:\Omega\times \Omega\longrightarrow\mathbb{R}$, $f(X_{1}(\omega_{1}),X_{2}(\omega_{2}))\mapsto X_{1}(\omega_{1})+X_{2}(\omega_{2})+37$ we see that $f$ is $\mathscr{F}/\mathscr{R}$ measurable - i.e. it is a random variable. Furthermore the probability space is again $(\Omega,\mathscr{F})$, so nothing new (probabilistically) is needed. Other properties aside from continuity also give measurable mappings, but the main point is that it will not do to take any function.
Another way of viewing all this is as the composition of two functions $f\circ W$, for $W=(X_{1},X_{2}):\Omega\times\Omega\longrightarrow\mathbb{R}\times\mathbb{R}$. We have $W$ being $\mathscr{F}/(\mathscr{R}\times \mathscr{R})$ measurable and $f$ being $(\mathscr{R}\times \mathscr{R})\times\mathscr{R}$ measurable. There is a result that states in these circumstnces that $f\circ W$ is then $\mathscr{F}/\mathscr{R}$ measurable
To your likelihood question now. Let the joint density for $W=(X_{1},X_{2})$ be denoted $f(y,z|\theta)$ for parameter $\theta\in\Theta$. By definition $W$ is $\mathscr{F}/(\mathscr{R}\times \mathscr{R})$ measurable (since $W$ is a random variable) and $f:\mathbb{R}\times \mathbb{R}\longrightarrow \mathbb{R}$ is $(\mathscr{R}\times \mathscr{R})\times\mathscr{R}$ measurable (since $f$ is the density for the distribution of $W$) so as above $f\circ W\mapsto f(X_{1}(\omega_{1}),X_{2}(\omega_{2})|\theta)$ is then $\mathscr{F}/\mathscr{R}$ measurable and $Pr[f(X_{1},X_{2}|\theta)\in B]=P\left\{[f(X_{1},X_{2}|\theta)]^{-1}(B)\right\}$ for all $B\in\mathscr{R}$. So again the original probability measure $P$ and $\sigma$-field $\mathscr{F}$ are enough to describe the randomness in $f$.
Edit: posted an incomplete answer since OP just removed this from maths stackexchange – I will now try and finish before this is removed again!
Continuing
It is worth mentioning the relationship between the distribution (law) for $W$ and $f$ the density. Let $\mu_{\theta}$ be the probability law for $(X_{1},X_{2})$ for a $\theta\in\Theta$ then by definition $Pr[(X_{1},X_{2})\in R|\theta]=\mu_{\theta}(R)=\int_{R}f(y,z|\theta)d(y,z)$ for all $R\in\mathscr{R}\times\mathscr{R}$ (this is the $\sigma$-field of subsets of $\mathbb{R}\times\mathbb{R}$) which shows that probabilities of $f$ being in the linear Borel sets $\mathscr{R}$, i.e $Pr[f(X_{1},X_{2}|\theta)\in B]$ are not the same concept as the probabilities of $W$ being in the $2$-dimensional Borel sets $\mathscr{R}\times\mathscr{R}$, i.e $Pr[(X_{1},X_{2})\in R|\theta]$. Again $f(X_{1},X_{2}|\theta)$ is just a random function to which $P$ can assign probabilities to (of $f$ being in certain sets).
The likelihood function $L(\theta|x_{1},x_{2})$ is, as you say, is just $f(x_{1},x_{2}|\theta)$ viewed as a function of $\theta\in\Theta$ rather than $W\in\mathscr{R}\times \mathscr{R}$. Nonetheless this "viewpoint" is for one realisation of $W=w$. When inference is performed we think of repeated realisations of $W$ (in a frequentist inference setting anyway), and so we write $L(\theta|X_{1},X_{2})=f(X_{1},X_{2}|\theta)$. For fixed $\theta$ this is exactly as stated above - a $\mathscr{F}/\mathscr{R}$ measurable function. Thus for $\theta',\theta''\in\Theta$, $L(\theta'|X_{1},X_{2})$ and $L(\theta''|X_{1},X_{2})$ are both $\mathscr{F}/\mathscr{R}$ measurable functions that can be described, probabilistically, with $P$ and $\mathscr{F}$, so no new probability space needed.
As answered above by another user, the key difference to $f$ is that the analagous result to
$Pr[(X_{1},X_{2})\in \mathbb{R}^{2}|\theta]=\mu_{\theta}(\mathbb{R}^{2})=\int_{\mathbb{R}^{2}}f(y,z|\theta)d(y,z)=1$
does not hold for $L$, that is for any observed $x_{1},x_{2}$
$\int_{\Theta}L(\theta|x_{1},x_{2})d(\theta)\not=1$
