A random variable $X$ with differentiable distribution function has a density

Question

Setting: My professor defined

A random variable $X: \Omega \to \mathbb{R}$ has a density $f:\mathbb{R} \to \mathbb{R}$ if for all $B \in \mathscr{B}$ $$P(X^{-1} (B)) = \int_\mathbb{R} 1_{B}(\lambda) f(\lambda) d\lambda.$$

Here $\mathscr B$ denotes the Borel-$\sigma$-Algebra on $\mathbb{R}$.

My Problem: I have to prove that a random variable $X : \Omega \to \mathbb{R}$ with continuously differentiable distribution function $F$ has a density $f$.

What I did so far: Since $F$ is continuously differentiable, I set $f:=F'$. Then $$ \int_\mathbb{R} 1_{(-\infty,c]} f(\lambda) d\lambda=\int_{-\infty}^cf(\lambda)d\lambda = F(c)-\lim_{c \to -\infty} F(c) = F(c) - \lim_{c \to -\infty} P(X\leq c)=F(c) = P(X^{-1}((-\infty,c]))$$ which shows the statement for sets of the form $B=(-\infty,c]$.

Where I failed: I can't show that this also holds for general $B \in \mathscr{B}$. I know that the sets $(-\infty,c]$ constitute a basis for the Borel-$\sigma$-Algebra but I don't know how to generalize the proof to more general Borel sets.

Can someone give me a just a hint on how to start? Any help is much appreciated!

P.S. I know that most books define "density" only by means of the sets $(-\infty,c]$ but my professor did not and I need to use his definitions.

In your definition, don't you mean $1_B(\lambda)$ instead of $1_{(c,+\infty]}(\lambda)$? — Ian, May 16 '14 at 13:39
In your professor's definition, what, if any, is the relationship of $c$ to the Borel set $B$? — Dilip Sarwate, May 16 '14 at 13:39
Sorry, that was a typo. It says $1_B$. I changed it in the post. — mjb, May 16 '14 at 13:49

score 3 · Accepted Answer · answered May 16 '14 at 13:49

You said that you know that $\sigma\left((-\infty,c]\ :\ c\in\mathbb R\right)=\mathscr B$.

Set $\mathbb Q(B)=\int_Bf(\lambda)\,\mathrm d\lambda$ for $B\in\mathscr B$. You have already proved that $\mathbb P\circ X^{-1}$ and $\mathbb Q$ coincide on the sets $(-\infty,c]$ which generate $\mathscr B$.

Define the set $$ \mathcal M=\left\{B\in\mathscr B\ :\ \mathbb P\circ X^{-1}(B)=\mathbb Q(B)\right\}\supset\left\{(-\infty,c]\ :\ c\in\mathbb R\right\}, $$ and check that $\mathcal M$ is a monotone class. Then, conclude by the monotone class theorem.

Thanks a lot! That does the trick – mjb May 16 '14 at 16:10 — mjb, May 16 '14 at 16:10

score 1 · Answer 2 · answered May 16 '14 at 14:49

1

This is a matter of notation. Let $B_c=(-\infty, c)$ be an open set. Then we have

$$ P(X^{-1}(B_c))=\int^{c}_{-\infty}f(\lambda)d\lambda $$ Note that $X^{-1}(B_c)=\{\omega:X(\omega)<c\}$. Therefore $P(X^{-1}(B_c))=F_{X}(c)$. The rest just follows.

answered May 16 '14 at 14:49

Bombyx mori

20,152

Thank you but I disagree. How "the rest just follows" is precisely my question. I do already know that the equality holds for sets of this type. I need to know it for general Borel sets. – mjb May 16 '14 at 16:11
@mjb: If my memory does not fail me, for any Borel set you can use a $G_{\delta}$ type set to approximate it such that the difference has zero measure. Then you proved the statement for all Borel sets. – Bombyx mori May 16 '14 at 18:19

A random variable $X$ with differentiable distribution function has a density

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