It turns out that yesterday’s churning media rumor mill was right about presidential candidate Rudy Giuliani picking up a big endorsement Friday, it was just wrong about the endorser. The AP is reporting that former Wisconsin Gov. Tommy Thompson, who ended his own White House bid in August, will officially announce his support for Giuliani today in Charleston.

”Rudy Giuliani has shown that he is a true leader. He can and will win the nomination and the presidency. He is America’s mayor, and during a period of time of great stress for this country he showed tremendous leadership,” Thompson, the former Wisconsin governor, said in a statement obtained by the Associated Press. …

Thompson is the first former 2008 Republican presidential candidate to endorse in the race, and officials familiar with details of the political courting involved said Giuliani’s top rivals also sought his support. But Thompson recently told Giuliani he’d get the nod after several weeks of conversations.

Thompson, the Health and Human Services secretary in President Bush’s first term, entered the crowded GOP presidential field early this year only to spend several months struggling to raise money and gain traction in early voting Iowa, New Hampshire and elsewhere. He was overshadowed by another, unrelated Thompson who was not even in the race at the time, but who was polling far stronger than the ex-governor, former Tennessee senator Fred Thompson.

The lesser-known Thompson dropped out of the race in August after finishing sixth in an Iowa test vote measuring organization and popularity in the state that holds the leadoff caucuses.

Unclear is the impact of Thompson’s support for Giuliani. On one hand, Giuliani could use the endorsement to try to mollify skeptical Iowans and other Midwesterners who are concerned about his moderate-to-liberal stances on social issues, pointing out that a prominent Republican from their backyards is supporting him.

On the other hand, Thompson left the race because voters weren’t gravitating toward him and he doesn’t bring with him much support in early voting Iowa and elsewhere. He languished in single digits in polls there, and barely registered in national surveys while in the race. [AP]