Related to Cauchy-random variables why is the Lebesgue-integral of the above (expectation) of such r.v not defined?
$$\int_{\mathbb{R}} \frac{x}{\pi(1 + x^2)} = ?$$
It seems to me that in the construction of the step functions, I want to argue by symmetry of a factor of $-1$ that this is $0$. But, the expectation is undefined which I can see when evaluating the Riemann integral as I know how. However, I do not have any measure theory background and am confused by the question of the Lebesgue integrability of such a function.
A function is in $L^1$ if both the positive part and the negative part of the function, $f_-=\max\{-f,0\}$ and $f_+=\max\{f,0\}$ (both are positive) have a finite integral. Then you define $$ \int f\,d\mu=\int f_+\,d\mu-\int f_-\,d\mu $$ which is well-defined (finite). In your case, both of these are infinite, so your distribution does not have a first moment (expected value).