What is the ruling on assisting neighbors in their innovations (such as the Great or Small Thursday, or the forty-day commemoration upon a death) by supplying electricity or fetching things, and is that considered aiding in sin?

Observing the "Arba'een" (fortieth day) and "Khamis" (Thursday rituals) is considered an impermissible innovation (bid'ah). Therefore, assisting in them is not permissible, as it falls under aiding in sin and transgression. To avoid problems with neighbors, one can help the deceased's family with permissible matters such as washing the body, shrouding it, digging the grave, and preparing food, then apologize for the innovations. It is appropriate to remind them that the deceased does not benefit from these innovations, and that ongoing charity (sadaqah jariyah) is more beneficial for them.