# Law of a step function and its generalization to two dimensions on an appropriate spaces

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

• A step function: $$u_1(x)=\begin{cases} u_{L}, xc_1, \end{cases}$$
• A "generalization to two dimensions": $$u_2(x,t)=\begin{cases} u_{L}, xc_2\cdot t. \end{cases}$$

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.

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 $$\tilde u_2$$.

• Thank you for the answer. For my problem it doesn't important what $u_1(c_1)$, $u_2(c_2 \cdot t, t)$ is. You could take $u_R$ for both of them. My guesses for $u_1$ would be $L^{\infty}(D)$ too, but also $BV(D)$ - space of functions with bounded variation, $\mathcal{M}(D)$ - space of Radon measures and $\mathcal{D}^{'}(D)$ - space of distributions. I am not sure if I am right. For the function $u_2$ the space $L^{\infty}(D \times [0,T])$ sounds good. I guess also that the spaces $BV(D \times[0,T])$, $\mathcal{M}(D \times [0,T])$, $\mathcal{D}^{'}(D \times [0,T])$ could work too? – Mark Mar 9 '20 at 18:10
• @Mark : Yes, you can use many other spaces instead of $L^\infty$. However, for $BV$ and $\mathcal D'$, you need to assume that $D$ is open and also replace $[0,T]$ by $(0,T)$. – Iosif Pinelis Mar 9 '20 at 18:18
• Thanks for the correction on $BV$ and $\mathcal{D}^{'}$. For $u_1$ spaces such as $\mathcal{M}$ and $L^p, p\in[0,\infty]$ sounds good for application. However I am not sure for the $u_2$. For it I need some Banach space-valued spaces. Last year I asked question on MSE for the function $u_2$ (edited today) (math.stackexchange.com/questions/3254479/…). There is a big list there at the end of it. I think that at least some of those Banach space-valued spaces could work. Am I wrong? And thanks again. – Mark Mar 9 '20 at 18:33
• @Mark : I have added a response to your latest comment. – Iosif Pinelis Mar 9 '20 at 22:05
• @LSpice, thanks for the correction. It is one of those things I didn't learn correctly from the start, and now I make that mistake from time to time. – Mark Mar 10 '20 at 9:50