For some random variable $X\in R^d$, let $m(\lambda) = E[\exp\langle\lambda,X\rangle]$ be the moment generating function. Now define a probability measure $$\mu_\lambda(S) = m(\lambda)^{-1}E\left[\exp\langle\lambda,Y\rangle \, 1_{Y\in S}\right].$$ Usually we'll be interested in $\mu_{\lambda^*}$ where $\lambda^*=\arg\min_\lambda m(\lambda)$.
I have seen this in texts on sharp large deviations, such as "The Probability in the Extreme Tail of a Convolution " - https://projecteuclid.org/euclid.aoms/1177706094 , and it seems very convenient. I don't have much intuition for it though, and would like to know its name so I can read more about it.
Edit: I realize now that this is the Cramer Transform. Not to be confused with the Lagrange Transform used in Cramer's Theorem. In Iltis 92 it is also called "the v-centered conjugate or "twisted" distribution, (or "Cramer transform")".
I still wonder if anyone has written a good overview of its history and properties.