Is the expiation for an oath (kaffarat al-yamin) incumbent upon one who swore to inform his brother of his wife's mistake, and then did not do so?

You did well in retracting your intention to inform your brother about your wife's words. However, you must offer expiation, for the Prophet, peace and blessings be upon him, said: "Whoever takes an oath and then sees something else as better than it, let him do what is better and offer expiation for his oath."