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Let $X$ be a random variable on a probability space $(\Omega, \mathcal{F}, P)$ and let $g$ be a Borel-measurable function on $\mathbb{R}$. In Shreve II (p 28) he proves, using the standard machine, that $$ \mathbb{E}(g(X)) = \int_{\mathbb{R}} g(x)\, d(P \circ X^{-1})(x), $$ where $P \circ X^{-1}$ is the pushforward measure on $\mathbb{R}$. He then again uses the standard machine to prove that, for a continuous random variable $X$, that $$ \mathbb{E}(g(X)) = \int_{\mathbb{R}} g(x) f(x) d\lambda(x), $$ where $f$ is the probability density function of $X$ and $\lambda$ is the Lebesgue measure on $\mathbb{R}$.

My question is, is the standard machine really necessary for this second part? By definition of a continuous random variable, $f = \frac{d(P \circ X^{-1})}{d \lambda}$, and so $$ \int_{\mathbb{R}} g(x)\, d(P \circ X^{-1})(x) = \int_{\mathbb{R}} g(x)f(x) d\lambda(x) $$ since $f$ is the Radon-Nikodym derivative of $P \circ X^{-1}$ w.r.t. $\lambda$.

Perhaps I am overlooking some integrability conditions?

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As the Radon-Nykodim derivative is defined for measures on the same measurable space, and while probability $P$ is defined on the probability space $\Omega$, the standard machine is necessary to have the two measures $\lambda$ and $P \circ X^{-1}$ both defined on $\mathbb{R}$. That is, \begin{align*} \mathbb E(g(X)) &=\lim_{n\rightarrow \infty}\sum_{m=-\infty}^{\infty}\sum_{i=1}^{2^n}\Big(\frac{i-1}{2^n}+m\Big)P\Big(\frac{i-1}{2^n}+m \leq g(X) < \frac{i}{2^n}+m\Big)\\ &=\lim_{n\rightarrow \infty}\sum_{m=-\infty}^{\infty}\sum_{i=1}^{2^n}\Big(\frac{i-1}{2^n}+m\Big)P \circ X^{-1}\bigg(g^{-1}\Big[\frac{i-1}{2^n}+m, \ \frac{i}{2^n}+m\Big)\bigg)\\ &=\int_{\mathbb{R}} g(x)\, d(P \circ X^{-1})(x)\\ &= \int_{\mathbb{R}} g(x) f(x) d\lambda(x), \end{align*} where $f = \frac{d(P \circ X^{-1})}{d \lambda}$ is the Radon Nykodim derivative.

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  • $\begingroup$ the two measures $P \circ X^{-1}$ and $\lambda$ are both defined on $(\mathbb{R}, \mathcal{B})$, which is why I believe (correct me if I'm wrong!) the Radon-Nikodym $f$ above is valid. $\endgroup$ – bcf Jul 2 '15 at 16:12
  • $\begingroup$ @bcf, that is correct. $\endgroup$ – Gordon Jul 2 '15 at 16:12
  • $\begingroup$ Okay, thanks. I see that the standard machine is necessary for my first equation above, but I'm wondering if it's necessary to show the third equation, or if we can just use the definition of the Radon-Nikodym derivative? $\endgroup$ – bcf Jul 2 '15 at 16:58
  • $\begingroup$ @bcf, you can show the third equation from the definition of Radon Nykodim derivative, however, it is only a one-line proof, as for any Borel set $B$, $P \circ X^{-1}(B) = \int_B f(x) d\lambda$. $\endgroup$ – Gordon Jul 2 '15 at 17:08

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