ASHINGTON, Nov. 1 - The Supreme Court agreed Monday to decide an important case about the rights of those who turn to the police for protection against a violent family member or acquaintance.

The question, framed in a lawsuit brought by a woman whose estranged husband murdered their three young daughters, is whether the failure by the police to enforce a protective court order she had obtained violated her constitutional right to due process.

The lower courts have confronted the question numerous times in recent years and have produced conflicting answers. In the case the justices accepted, the United States Court of Appeals for the 10th Circuit, in Denver, voted 6 to 5 to permit the woman's lawsuit, against the town of Castle Rock, Colo., to proceed to trial. The federal district court in Denver had dismissed the suit, in which the plaintiff, Jessica Gonzales, is seeking $30 million in compensatory damages and millions more in punitive damages.

Colorado law makes it mandatory for the police to provide protection once a protective order is issued. "A peace officer shall use every reasonable means to enforce a protection order," the state law provides. In addition, the law requires the police to make an arrest if they have probable cause to believe that such an order has been violated. Many other states have similar laws.

The appeals court's majority reasoned that the law gave Colorado residents a vested right to receive a particular government service, police protection, that cannot be withheld without procedural safeguards intended to prevent the government from acting in an arbitrary way.

The case will require the Supreme Court to revisit a doctrine articulated by Chief Justice William H. Rehnquist in a well-known case from 1989, in which a county social services department in Wisconsin was found not to have breached a constitutional duty when it returned a young boy to an abusive father and then failed to monitor his safety. That decision, DeShaney v. Winnebago County, established the rule that the government is not ordinarily obliged to protect people from harm at the hands of their fellow private citizens.

The DeShaney opinion was based on a branch of constitutional analysis known as "substantive due process." The 14th Amendment bars the states from depriving "any person of life, liberty, or property" without due process, and the substantive due process question asks whether the challenged government action has violated any of those three guarantees.

"Nothing in the language of the Due Process Clause itself requires the State to protect the life, liberty, and property of its citizens against invasion by private actors," Chief Justice Rehnquist said in his opinion.

The 14th Amendment also has a procedural component: even if the state has a permissible reason for impinging on the amendment's substantive protections, it must use due process proper procedures in doing so. In the DeShaney case, Chief Justice Rehnquist said the question of whether the county had used proper procedures in its care for the child had not been properly presented to the court. Consequently, the ruling has been interpreted by some lower courts as not foreclosing a procedural ruling in a future case.

That was the approach the 10th Circuit took in the new case, Town of Castle Rock v. Gonzales, No. 04-278. The majority said Ms. Gonzales had presented a plausible case that the police had failed to follow proper procedure in disregarding her repeated requests for help after her husband had kidnapped the three girls - ages 7, 9 and 10 - from the yard of their house on the afternoon of June 22, 1999.

Ms. Gonzales reported the girls missing, telling the officers who arrived at her house that she suspected her husband and feared that he might be violent. She showed the officers a copy of the restraining order she had obtained a month earlier in connection with their divorce case.

But the police failed to take any action, even when Ms. Gonzales learned that her husband, Simon Gonzales, was with the girls at a local amusement park. She called the police repeatedly and went to the police station herself after midnight. At 3:20 in the morning, Mr. Gonzales arrived at the police station in a truck, opened fire with a semiautomatic handgun he had bought the previous day and was shot dead at the scene. The bodies of the three girls were in the cab of the truck.

In its Supreme Court appeal, the town's lawyer argued that by permitting the lawsuit to proceed, the 10th Circuit's decision threatened to "convert hundreds of procedural mandates into constitutional claims" and could "bankrupt" municipal governments for their inevitable instances of "less than perfect" law enforcement.

"Federal courts should not lay such a heavy hand on evolving state efforts to address such quintessentially local issues of crime and violence," the brief said.

The town is supported in its appeal by the National League of Cities and the International Municipal Lawyers Association.

The justices also announced Monday that the court would hear arguments in a territorial dispute between Alaska and the federal government over the ownership of underwater land within the boundaries of the Tongass National Forest and the Glacier Bay National Park and Preserve. A special master appointed by the court four years ago, Gregory E. Maggs, recommended awarding the disputed territory to the United States. The court will hear Alaska's appeal in Alaska v. United States, No. 128 Original.