Let's consider two discontinuous functions defined on $D$ and $D \times [0,T]$, respectively:


* A step function: $u\_1(x)=\begin{cases}
 u\_{L}, x<c\_1, \\[2ex]
 u\_{R}, x>c\_1,
 \end{cases}$
* A "generalization to two dimensions": $u\_2(x,t)=\begin{cases}
 u\_{L}, x<c\_2\cdot t, \\[2ex]
 u\_{R}, x>c\_2\cdot t.
 \end{cases}$


Here $x \in D \subseteq \mathbb{R}$, $t \in [0,T]$ and $c\_1,c\_2,u\_{L},u\_{R}$ are constants. Additionaly in the book "Stochastic equations in infinite dimensions, Da Prato G., Zabczyk J., 1992", we could find the definition of the *law*:


If $X$ is a random variable from $(\Omega,\mathcal{F})$ to $(E,\mathcal{S})$ and $P$ a probability measure on $\Omega$, then by $\mathcal{L}(X)$ we will denote the image of $P$ under the mapping $X$: $\mathcal{L}(X)(A)=P\{\omega \in \Omega:x(\omega)\in A\},\forall A\in \mathcal{S}.$ The measure $ \mathcal{L} (X)$ is called the *distribution* or the *law* of $X$. 


**As far as I know the probability law of $u\_1$ and $u\_2$ is the Dirac mass at $u\_1$ and $u\_2$, respectively, seen as a measure on an appropriate function space.** Although I can't remember where I have read this in the literature. If my recollection is wrong, please correct me.


If we assume that the last paragraph above is correct, my question is: **What would be those appropriate spaces in the cases of the functions $u\_1$ and $u\_2$?** 


Example I got last week for the $u\_1$: the law of the function $u\_1$ is the Dirac measure concentrated at $u\_1$ on the space of cadlag function from [0,1] to $\mathbb{R}$. But for the problem I have in my mind cadlag functions probably won't work. 


For the function $u\_2$ I don't have any examples. I think that the space of $BV$ functions from $(D\times [0,T])$ to $\mathbb{R}$ should be one of the appropriate spaces. But my ideal appropriate spaces should look as $C([0,T];\mathcal{M}(D))$ or similar - they should be Banach space-valued. 


I work on one problem for a few weeks that concerns these two functions and in order to apply the technique that was recommended me (in order to solve it), I need to consider the laws of this two functions. I need help with this. Any appropriate space you recommend me is welcome. Thanks everyone in advance.



Law of a step function and its generalization to two dimensions on an appropriate spaces
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](https://en.wikipedia.org/wiki/Dirac_measure) on $B\_i$ at $u\_i$. 




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**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 $\tilde u\_2$. 


