New York City has won a victory on the insurance costs of defending against as many as 10,000 claims of respiratory and other illness by construction workers, police officers, firefighters and others who responded to the catastrophe on Sept. 11, 2001.

Southern District of New York Judge Alvin K. Hellerstein has ruled that Lloyd’s of London and other insurers are responsible for $100 million in defense costs already incurred and what could be at least twice that amount in future costs.