Compact support functions and law of a random variable

[psie]

TL;DR Summary

I don't understand the characterization of the law of a random variable through continuous compact support functions. In particular, I don't understand why we need to consider the whole set of all these functions rather than a subset.

I'm reading in my probability book about characterizations of the law of a random variable, that is, the probability measure $\mathbb P_X(A)=\mathbb P(X\in A)$. I read the following passage (I'm paraphrasing slightly):

Since the indicator function of an open rectangle (i.e. $(a_1,b_1)\times (a_2,b_2)\times\cdots\times (a_n,b_n)$, for any choice of the reals $a_1<b_1,\ldots,a_n<b_n$) is the increasing limit of a sequence of continuous functions with compact support, the probability measure $\mu$ is also determined by the values of $\int \varphi\,\mu(\mathrm{d}x)$ when $\varphi$ varies in the space $C_c(\mathbb R^n)$ of all continuous functions with compact support from $\mathbb R^n$ into $\mathbb R$.

This extract is basically saying that if $$\mathbb E[\varphi(X)]:=\int \varphi\,\mathbb P_X(\mathrm{d}x)=\int \varphi\,\mathbb P_Y(\mathrm{d}x)=:\mathbb E[\varphi(Y)],\quad \varphi\in C_c(\mathbb R^n),$$then $\mathbb P_X=\mathbb P_Y$. Indeed, if $U$ denotes an open rectangle and we define $$f_n(x)=\min\{n\cdot d(x,U^c),1\},\quad n\geq 1,$$then $f_n$ are continuous and have compact support (also, they are nonnegative), since the closure of $U$ is a closed rectangle, which is compact in $\mathbb R^n$. And $f_n\nearrow \mathbf1_U$ as $n\to\infty$, so by monotone convergence, $$\mathbb P_X(U)=\int \mathbf1_U\,\mathrm d\mathbb P_X=\lim_{n\to\infty}\int f_n\,\mathrm{d}\mathbb P_X=\lim_{n\to\infty}\int f_n\,\mathrm d \mathbb P_Y=\mathbb P_Y(U).$$ So $\mathbb P_X=\mathbb P_Y$ for all open rectangles, and by an argument one can make via the $\pi-\lambda$ theorem, they are equal.

Now, what troubles me is that, why does the author of the text stipulate that $\varphi$ needs to vary over all functions in $C_c(\mathbb R^d)$? Isn't it enough, as I've shown, to just take the nonnegative continuous functions with compact support?

 

[psie]

Maybe I'm overthinking things. If we have the information $\mathbb E[\varphi(X)], \varphi\in C_c(\mathbb R^n)$, then we also have $\mathbb E[\phi(X)]$ for $\phi$ being a nonnegative, continuous function with compact support on $\mathbb R^n$.

I guess what I was after was making the statement slightly weaker by demanding less; the nonnegative continuous functions with compact support on $\mathbb R^n$ are clearly a subset to $C_c(\mathbb R^n)$.

 

[fresh_42]

This isn't Durrett anymore, isn't it? It would help to see the entire context and not just your personal view which might already be contaminated by your "overthinking".

 

[psie]

No, it is Measure Theory, Probability and Stochastic Processes by Le Gall. I quote the whole relevant passage, as it is stated in the book. This passage appears in a section on expectation, in particular after a proposition that states that expectation of a function of random variable taking values in $E$ is simply $\mathbb E[f(X)]=\int_E f(x)\,\mathbb P_X(\mathrm{d}x)$, i.e. the law of the unconscious statistician.

Characterization of the Law of a Random Variable It will be useful to characterize the law of a random variable in different ways. It follows from Corollory 1.19 that a probability measure $\mu$ on $\mathbb R^d$ is characterized by its values on open rectangles, that is, by the quantities $\mu((a_1,b_1)\times (a_2,b_2)\times\cdots\times (a_d,b_d))$, for any choice of the reals $a_1<b_1,\ldots,a_n<b_n$ (one could restrict to rational values of $a_i,b_i$, and/or replace open intervals by closed intervals). Since the indicator function of an open rectangle is the increasing limit of a sequence of continuous functions with compact support, the probability measure $\mu$ is also determined by the values of $\int \varphi\,\mu(\mathrm{d}x)$ when $\varphi$ varies in the space $C_c(\mathbb R^d)$ of all continuous functions with compact support from $\mathbb R^d$ into $\mathbb R$.

Corollary 1.19 reads (only part (i) is relevant, as that part applies to finite measures):

Corollary 1.19 Let $\mu$ and $\nu$ be two measures on $(E,\mathcal{A})$. Suppose that there exists a class $\mathcal{C}\subset\mathcal{A}$, which is closed under finite intersections, such that $\sigma(\mathcal{C})=\mathcal{A}$ and $\mu(A)=\nu(A)$ for every $A\in\mathcal{C}$.

(i) If $\mu(E)=\nu(E)<\infty$, then we have $\mu=\nu$.
(ii) If there exists an increasing sequence $(E_n)_{n\in\mathbb N}$ of elements of $\mathcal{C}$ such that $E=\bigcup_{n\in\mathbb N}E_n$ and $\mu(E_n)=\nu(E_n)<\infty$ for every $n\in\mathbb N$, then $\mu=\nu$.

 

[fresh_42]

Well, that's - other than Durrett - copyright protected so I cannot look it up and I'm not an expert in the field. Isn't the whole issue about exhausting an open set by a sequence of compact (and therefore measurable) subsets to get a reasonable definition for the probability for open, bounded sets? I'm not sure I get your point.

 

[psie]

Well, I'm not sure I understand my worries either, but basically I'm reacting to the statement that a probability measure $\mu$ is determined by the values of $\int \varphi\,\mu(\mathrm{d}x)$ when $\varphi$ varies in $C_c(\mathbb R^d)$. That integral could be negative, but a probability measure outputs only positive values, so I don't see why we'd be interested in varying $\varphi$ in all of $C_c(\mathbb R^d)$. I think there are functions in $C_c(\mathbb R^d)$ that are not part of any sequence that converge to an indicator function of an open rectangle.

Regarding exhausting an open set by a sequence of compact subsets, I was looking at these exercises here earlier, in particular problem 1(a). I think that is kind of what you meant, but they use the Lebesgue measure.

 

[fresh_42]

Well, I'm not sure I understand my worries either, but basically I'm reacting to the statement that a probability measure $\mu$ is determined by the values of $\int \varphi\,\mu(\mathrm{d}x)$ when $\varphi$ varies in $C_c(\mathbb R^d)$. That integral could be negative, but a probability measure outputs only positive values, so I don't see why we'd be interested in varying $\varphi$ in all of $C_c(\mathbb R^d)$. I think there are functions in $C_c(\mathbb R^d)$ that are not part of any sequence that converge to an indicator function of an open rectangle.

Regarding exhausting an open set by a sequence of compact subsets, I was looking at these exercises here earlier, in particular problem 1(a). I think that is kind of what you meant, but they use the Lebesgue measure.

Well, that would need a closer look at how these notations are defined. An integral is an oriented volume, so you can always get negative values, simply by the fact $\int_a^b +\int_b^a =0.$ That means you have to solve this system immanent problem anyway if you identify a probability with an integral, i.e. a volume.

 

[jbergman]

There is an exercise on p. 122 Rosenthal has an exercise to prove the equivalence of 6 different conditions for measures being equivalent including the ones you mentioned.

I think the importance is that different characterizations will make proving different theorems easier.

 

[WWGD]

Hmm, a technical point I may have missed/misunderstood: Given the indicator is $1$ within the open rectangles and $0$ elsewhere, it seems its support is the union of open rectangles and thus not compact. Am I missing something?

 

[jbergman]

Hmm, a technical point I may have missed/misunderstood: Given the indicator is $1$ within the open rectangles and $0$ elsewhere, it seems its support is the union of open rectangles and thus not compact. Am I missing something?

https://math.stackexchange.com/ques...hbbrn-is-the-increasing-union-of-compact-sets

 

[WWGD]

https://math.stackexchange.com/ques...hbbrn-is-the-increasing-union-of-compact-sets

Thanks. Isn't that property called $\sigma$-compactness or similar?

 

[WWGD]

Well, I'm not sure I understand my worries either, but basically I'm reacting to the statement that a probability measure $\mu$ is determined by the values of $\int \varphi\,\mu(\mathrm{d}x)$ when $\varphi$ varies in $C_c(\mathbb R^d)$. That integral could be negative, but a probability measure outputs only positive values, so I don't see why we'd be interested in varying $\varphi$ in all of $C_c(\mathbb R^d)$. I think there are functions in $C_c(\mathbb R^d)$ that are not part of any sequence that converge to an indicator function of an open rectangle.

Regarding exhausting an open set by a sequence of compact subsets, I was looking at these exercises here earlier, in particular problem 1(a). I think that is kind of what you meant, but they use the Lebesgue measure.

There is such thing as signed measures.
https://en.m.wikipedia.org/wiki/Signed_measure