Surrounded by Rahim's brothers, mother and wife on the sidewalk where Rahim was shot, Sullivan said Thursday that they hope to work "productively and cooperatively" with investigators in a "joint effort to search for the truth."
"Let me be clear: The family is not making any substantive claims at this time about what happened in this case," Sullivan said.
A Sullivan spokeswoman said in an email Friday that the family would not comment on the video.
Rahim, 26, was fatally shot Tuesday after investigators said he refused to drop a military-style knife as they sought to question him about "terrorist-related information." Police have said the video shows officers backing up and Rahim moving toward them before they fired.
Sullivan said Rahim's brother Ibrahim Rahim, a well-known imam in Boston, regrets posting a Facebook message based on incorrect, third-hand information claiming that his brother had been shot three times in the back.
"We now know we simply did not have all the facts at that time," Sullivan said.
Police Commissioner William Evans said Rahim had talked about beheading blogger Pamela Geller before deciding to target police officers. "There was some mention of that name," said Evans, who dismissed the idea as "wishful thinking" while speaking on the "Today" show.
A Muslim leader said Thursday that his killing by Boston police and the FBI was reckless and unnecessary. Imam Abdullah Faaruuq, who is close to the Rahim family, said he believes they wanted him dead.
"You can capture elephants, lions and tigers without killing them," Faaruuq said. "The intent was not to capture him and keep him alive."
Rahim will be buried Friday, but the family is keeping it private.
Geller is a combative personality known for provoking Muslims by campaigning against a mosque near the World Trade Center site, sponsoring inflammatory advertisements and organizing Prophet Muhammad cartoon contest in Texas.
"They want to make an example out of me to frighten the rest of the US into silence and submission," Geller wrote in an email to The Associated Press on Thursday. "This is not about me. This is about whether the US will stand for freedom or submit and cower before violent intimidation."