Q: Strictly convex function measure Lebesgue - homework Let $\mu$ a probability measure over $\mathbb R$. Show that if for some strictly convex function $f: \mathbb R \rightarrow \mathbb R$ occurs
$$
\int_{\mathbb R} f(x)d\mu (x) = f(x_0).
$$
for some $x_0 \in \mathbb R$, then $\mu = \delta_{x_0}.$
I use the definition about strictly convex function, but I cannot get $\mu$
I try to use a function 
$$h(x) = \mu(x) - \delta_{x_0}(x)$$
nothing happened.
I get a inequality
$$
\int_{\mathbb R}f((1-\lambda)x + \lambda y)d\mu(x)  \leq (1- \lambda)f(x_0) + \lambda f(x_0).$$
Could you give me a hand?
