International applications of double jeopardy

A Japanese business, Kazuyoshi Miura, has been arrested for the 1981 murder of his wife in LA. He had been convicted in a Japanese court, but then won an appeal to the Tokyo High Court (of appeals). It did not get taken to the Supreme Court, and he was freed after 13 years in prison for a separate count of assaulting his wife previous to the murder.

My question:

He was convicted in Japan of the murder in America, and then acquitted on appeal. He can’t be tried again in Japan, because of double jeopardy, but since the LAPD has had him arrested, obviously they believe they can try him in the States.

Does double jeopardy not apply to cases where defendants are tried for the same offence in different countries?

To get you started, from here: Double jeopardy - Wikipedia