Consider probability density functions $f_{1}\left(x\right)$ and $f_{2}\left(x\right)$ and the mixture distribution $$f_{3}\left(x\right)\equiv pf_{1}\left(x\right)+\left(1-p\right)f_{2}\left(x\right)$$ Assume that the moments associated with $f_{1}\left(x\right)$ are all clearly defined, but the moments of $f_{2}\left(x\right)$ may not necessarily be (e.g. a Cauchy or Levy distribution).
To calculate the expected value one would find $$\int_{-\infty}^{\infty}xf_{3}\left(x\right)dx=p\int_{-\infty}^{\infty}xf_{1}\left(x\right)dx+\left(1-p\right)\int_{-\infty}^{\infty}xf_{2}\left(x\right)dx$$ Presumably if $\int_{-\infty}^{\infty}xf_{2}\left(x\right)dx$ is infinite or undefined, then the expected value only exists when $p=1$.
Is this logic correct, even if $p$ is close to but less than $1$? Does it extend to higher moments as well?