Nonetheless, Gov. Steve Beshear is right to try to shut down Internet gambling sites that, by his estimate, attract “tens of millions of dollars” each year from Kentuckians.

Not insignificantly, those are untaxed dollars, and losses are directed out of the country rather than toward Kentucky businesses, banks and legal gambling activities, including racetracks. Moreover, since most Internet gambling is off-shore and unregulated, players have few if any assurances that the games are honest and that they will be paid if they win.

Previous American efforts to curb Internet gambling generally have been disappointing. It has flourished despite widespread state prohibitions, and states that have tried rigorous enforcement have had few positive results.

A 2006 federal law designed to thwart Internet gambling by forbidding banks and other financial institutions from transferring funds to and from illegal gambling operations is often circumvented by establishment of money-handling intermediaries.

Finally, the role of off-shore gambling operations in a global economy is complex. Indeed, complaints about U.S. restrictions have been brought to the World Trade Organization.

For all that, however, the Beshear administration’s initiative is intriguing. Kentucky’s law banning Internet and other unauthorized betting is strict, and the state can demonstrate genuine harm to racing and regulated gambling.

More important, the state’s strategy — pursuing a civil lawsuit to try to force Internet gambling sites to block access by Kentucky players or else surrender control of their domain names — seems novel.

One possible outcome is that Internet gambling operators might agree to regulation and taxes.

To have strong impact, even if successful in court, Kentucky’s course likely would need to be adopted by other, larger states or by the federal government.

But that’s possible. The case here is likely to be watched closely.