look through: 仔细查阅
Peter starts looking through the mail as soon as the door shuts.
彼得一关上门就开始逐一查看起邮件来。
He happened to be looking through the medical book 'Gray's Anatomy' at the time.
那时，他正好在看医书《格雷氏解剖学》。

double check: 仔细检查
I'll double-check the figures.
我将对这些数字进行复核。
Don'tbelieve what you are told; double-check with an independent source.
别相信别人告诉你的话，用独立的信息来源再核实一遍。