Your functions $u_1$ and $u_2$ are not completely defined. For instance, $u_1(c_1)$ is undefined.

If you do not need to distinguish functions differing only on a set of Lebesgue measure $0$, then you may consider $u_1$ a point in the Banach space $B_1:=L^\infty(D)$, and $u_2$ a point in the Banach space $B_2:=L^\infty(D\times[0,T])$, where $D$ is any nonempty Lebesgue-measurable subset of $\mathbb R$.

For $(\Omega,\mathcal F,P)$, you can take any probability space. For each $i\in\{1,2\}$, you can take any $\sigma$-algebra $\mathcal S_i$ over $B_i$, and then the probability law of the (nonrandom) random element $\Omega\ni\omega\mapsto u_i\in B_i$ will the Dirac measure on $B_i$ at $u_i$.

Your functions $u_1$ and $u_2$ are not completely defined. For instance, $u_1(c_1)$ is undefined.

If you do not need to distinguish functions differing only on a set of Lebesgue measure $0$, then you may consider $u_1$ a point in the Banach space $B_1:=L^\infty(D)$, and $u_2$ a point in the Banach space $B_2:=L^\infty(D\times[0,T])$, where $D$ is any nonempty Lebesgue-measurable subset of $\mathbb R$.

For $(\Omega,\mathcal F,P)$, you can take any probability space. For each $i\in\{1,2\}$, you can take any $\sigma$-algebra $\mathcal S_i$ over $B_i$, and then the probability law of the (nonrandom) random element $\Omega\ni\omega\mapsto u_i\in B_i$ will be the Dirac measure on $B_i$ at $u_i$.

Added in response to a comment by the OP: Of course, you can identify any function $w\colon X\times Y\to Z$ with the function $\tilde w\colon Y\to Z^X$ by the formula $\tilde w(y)(x):=w(x,y)$ for all $(x,y)\in X\times Y$.

In the case of your function $u_2\colon D\times[0,T]\to\mathbb R$, the corresponding function $\tilde u_2$ may be considered as follows: $$[0,T]\ni t\mapsto\tilde u_2\in \tilde B_2:=L^\infty\big([0,T],L^\infty(\mathbb R)\big),$$ where $D\ni x\mapsto\tilde u_2(t):=u_2(x,t)\in\mathbb R$. Then the probability law of the (nonrandom) random element $\Omega\ni\omega\mapsto\tilde u_2\in\tilde B_2$ will be the Dirac measure on the Banach space $\tilde B_2$ at the point $\tilde u_2$.

Your functions $u_1$ and $u_2$ are not completely defined. For instance, $u_1(c_1)$ is undefined.

If you do not need to distinguish functions differing only on a set of Lebesgue measure $0$, then you may consider $u_1$ a point in the Banach space $B_1:=L^\infty(D)$, and $u_2$ a point in the Banach space $B_2:=L^\infty(D\times[0,T])$, where $D$ is any nonempty Lebesgue-measurable subset of $\mathbb R$.

For $(\Omega,\mathcal F)$, you can take any measurable space. For each $i\in\{1,2\}$, you can take any $\sigma$-algebra $\mathcal S_i$ over $B_i$, and then the probability law of the (nonrandom) random element $\Omega\ni\omega\mapsto u_i\in B_i$ will the Dirac measure on $B_i$ at $u_i$.

Your functions $u_1$ and $u_2$ are not completely defined. For instance, $u_1(c_1)$ is undefined.

If you do not need to distinguish functions differing only on a set of Lebesgue measure $0$, then you may consider $u_1$ a point in the Banach space $B_1:=L^\infty(D)$, and $u_2$ a point in the Banach space $B_2:=L^\infty(D\times[0,T])$, where $D$ is any nonempty Lebesgue-measurable subset of $\mathbb R$.

For $(\Omega,\mathcal F,P)$, you can take any probability space. For each $i\in\{1,2\}$, you can take any $\sigma$-algebra $\mathcal S_i$ over $B_i$, and then the probability law of the (nonrandom) random element $\Omega\ni\omega\mapsto u_i\in B_i$ will the Dirac measure on $B_i$ at $u_i$.