Left hand radical (side-form of 手) — the indexing radical. Anchors 措 in the manual/managerial action family with 排 arrange, 推 push, 持 hold, 拥 embrace. Marks the entry as something done by the hands — practical placement, arrangement, handling.
Right 昔 supplies the sound: xī → cuò through historical sound shift. Same phonetic family: 错 wrong, 借 borrow, 惜 cherish. Pure sound marker; the 'former-days' meaning of 昔 (originally a flood image) doesn't carry into 措. Common in compounds like 措施 measures, 措辞 wording, 不知所措 at a loss.
