Here is something I do not understand for my lecture notes. The lemma is this.
Let $\mu$ be a probability measure on $R$, and $\Lambda^*_\mu$ is the Legendre transform of $\mu$. $\Lambda_\mu^*\geq 0$.
If $\int_R |x|\mu(dx)<\infty$ and $p=E(x) = \int_Rx\mu(dx)$, then $\Lambda^*_\mu(p)=0$. It is also non decreasing on $[p,\infty)$, non-increasing on $(-\infty,p]$
I worked through the proof, but got stuck when my lecturer set $p=\infty$. It says
$p=E(X_1)=-\infty$, the $\Lambda_\mu(\lambda)$, that is the log moment generating function $= \infty$ for $\lambda$ negative, $x\geq p$; $\lambda x-\Lambda_\mu(\lambda)\leq \Lambda^*_\mu(p)=0$
I have no clue what the last line is meant to say. It is not very clearly written. For $\lambda$ negative and $x\geq p$. I cannot see why that is true....