"Show that for the infinite "vocabulary" $\tau =\{P_1,P_2,\dots\} $ with unary $P_i$ there is an $L_{\omega_1\omega}^\omega[\tau]$-sentence without labeled asymptotic probability."
This is the Exercise 3.1.9 from Finite Model Theory by Ebbinghaus and Flum. This is kinda homework, however I'm stuck in this question for several days and nothing seems to work (of course, it is a counter-example). Can you give me any directions or hints?