There will be no recount of paper ballots in Pennsylvania, a federal judge ruled Monday.

US District Judge Paul Diamond rejected a request backed by Green Party presidential candidate Jill Stein to recount paper ballots and scan some counties' election systems for signs that the 2016 presidential election in Pennsylvania, where Donald Trump won by a narrow margin, was hacked.

In his 31-page decision, Judge Diamond wrote there existed at least six grounds that required him to reject the Green Party's lawsuit, writing that the suspicion that the election was hacked "borders on the irrational." The recount bid, he said, could "ensure that no Pennsylvania vote counts," as Tuesday is the federal deadline to certify the vote for the Electoral College.

"Most importantly," he wrote, "there is no credible evidence that any 'hack' occurred, and compelling evidence that Pennsylvania's voting system was not in any way compromised."

Dr. Stein, who finished fourth in the presidential election, had challenged the election results in Pennsylvania, Washington, and Michigan: three traditionally Democratic strongholds that supported President-elect Trump in the 2016 election. In Pennsylvania, she had argued, the state's use of electric voting machines with no paper trail in some districts meant that the system could have been hacked.

The decision to reject the Pennsylvania recount came the same day that Wisconsin election officials expected to complete a recount in their state. The Wisconsin recount was 95 percent complete as of Monday morning, and showed Trump with an increase of 628 votes, Democratic candidate Hillary Clinton with an increase of 653 votes, and Stein with an increase of 68 votes.

Stein's last-ditch appeal to secure a recount in Michigan was denied on Friday by the state's Supreme Court.

Elsewhere, some citizens have taken it upon themselves to demand recounts. In Florida, three voters have sued for a recount of all paper ballots, as Ellen Powell reported for The Christian Science Monitor last week:

Citing concerns about hacking, malfunctioning voting machines, and voters being turned away, three central Florida voters have brought a lawsuit against President-elect Donald Trump, Florida Gov. Rick Scott, and the Sunshine State’s 29 Republican electors. They are calling for a hand recount of all paper ballots, to be paid for by the defendants. The group’s lawyer, Clint Curtis, acknowledged that Mr. Trump, Governor Scott, and others can ignore the recount request entirely. But the suit, the latest of several to question the 2016 election results, may strengthen calls to address election issues.

Even if recounts had taken place in Wisconsin, Michigan, and Pennsylvania, it was considered extremely unlikely that they would have changed the outcome of the election to favor Mrs. Clinton, who also backed the recounts.

But some experts defended the effort, saying it could serve a valuable purpose in restoring confidence in the US electoral system and drive discussion about how to improve it.

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"Examining the physical evidence in these states – even if it finds nothing amiss – will help allay doubt and give voters justified confidence that the results are accurate," wrote J. Alex Halderman, a professor of computer science at the University of Michigan, for Medium. "It will also set a precedent for routinely examining paper ballots, which will provide an important deterrent against cyberattacks on future elections."

This report contains material from the Associated Press and Reuters.