Is the problem that determines whenever the word member $\in$ L(M) decidable or not?
I would try it like this:
Reduction from $A_{TM}$: Given TM $A$ und word $w$.
Use that to define a new machine $A'$ that works as follows:

Run like $A$ on input $w$
If $A$ rejects $w$, go into eternal loop.
If $A$ accepts, delete the band content and write &quot;member&quot;