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.