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I'm studying probability theory, and as a result measure theory, and I've come to the derivation for the expectation of a random variable in terms of its density function, i.e $$\mathbb{E}[X]:=\int Xd\mathbf{P}=\int_{\mathbb{R}}xf(x)dx$$

I've managed to understand pretty much everything in the derivation with the help of this question's accepted answer, but the one thing I don't understand is the evaluation at x. What I mean is, I understand using change of variables and Radon-Nikodym to get $$\int Xd\mathbf{P}=\int_{\mathbb{R}} \mathbf{id}_\mathbb{R} f d\lambda$$ but I fail to understand what it means to evaluate the integrand at $x$ to achieve $xf(x)d\lambda(x)$. Do we just write it as such for notations sake, like how $dx$ is used for $d\lambda(x)$, or does it mean something different than the integral of the functions themselves?

For instance, what does $d\lambda(x)$ even mean? From my understanding (I'm reading Achim Klenke's Probability Theory), $d\mu$ is used to represent the integral being "with respect to $\mu$", but $\lambda(x)$ (which would be 0 anyway, right?) is now a number not a measure. Does it mean $(d\lambda)(x)$? If so, is $d\lambda$ some new measure, and it doesn't just serve as an indication of what measure is being used for the integral?

modz
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  • Did you not learn $\int_a^b xf(x)dx$ in beginning calculus? If you have a specific function, this saves writing $\int_{[a,b]} g,d\mu$ and having to separately define $g[x)=xf(x)$. – Ted Shifrin Dec 08 '23 at 00:33
  • @TedShifrin I figured that for the functions it was just for convenience of notation (e.g writing $x^2$ instead of having to define a polynomial), but everywhere I look I see the same notation of $d\lambda(x)$ being used, which confused me into thinking something else was happening. Why not just leave it at $d\lambda$? – modz Dec 08 '23 at 01:00
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    To emphasize that $x$ is the variable of integration. Think about multiple integrals or integrating a function of more variables. – Ted Shifrin Dec 08 '23 at 01:05
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    There may be more than one variable involved, specifying disambiguates. Some even write $\lambda(dx)$ which has a pleasant 'physics-like' feeling to it. – copper.hat Dec 08 '23 at 03:09

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