Let $B \subseteq X$. Let $(X, \Sigma, \mu)$ be a measure space and let $c$ be the counting measure on $\mathcal P(\mathbb{N})$. Assume that $\{1\} \times B$ is measurable with respect to the product measure $c \times \mu$.
Is it true that $B$ is $\mu$-measurable?
I know that, in general, $A \times B$ being measurable does not necessarily imply $A$ and $B$ are. (For example, when $A= \emptyset$ and $B$ is not measurable). But if $A=\{1\}$ with the counting measure, I feel like it might work.
• $\mathcal P(\mathbb N)$ denotes the power set of $\mathbb N$