expression
my condolences; my deepest sympathy
A formal, polite set phrase used to express sympathy to someone who has lost a loved one. Often used in writing or in person at funerals.
この度はご愁傷様です。お悔やみ申し上げます。
I am very sorry for your loss. Please accept my deepest condolences.
お悔やみ申し上げますとともに、心よりご冥福をお祈りいたします。
I offer my condolences and pray for the repose of their soul.
Polite phrase combining お悔やみ (condolence) with 申し上げます (humble form of 言う, 'to say').