There is also a worry over whether the police and security services missed opportunities.
Since the attack, police have raised 12 properties in Barking and Ilford.
As many as seven people were killed and 48 others were injured. Police shot the attackers dead. One bedsit is believed to have been the attackers' safe house where the men plotted the horrific event.
In a rare glimpse into the weeklong investigation, police released details on Saturday that showed Khuram Butt originally tried to rent a 7.5 ton truck.
The Islamic gym where Butt worked out said Tuesday they had seen nothing of concern during his time there.
He added: "Just being around here and seeing everyone walking about, it's quite odd, because you can't imagine that anything so frightful might have actually happened".
The London Bridge attack ringleader Khuram Butt was considered a potential threat to British security and was still under active investigation at the time he carried out the deadly assault, United Kingdom counter-terrorism sources have told CNN.
He was credited with running toward the attackers and helping the injured rather than fleeing the area.
The family of 32-year-old James McMullan, who was killed in the rampage, issued a statement calling the attackers "deranged and deluded individuals".
Metropolitan Police said in a statement that Thomas was in town for the weekend with his girlfriend. Butt reportedly quoted the Koran in his final WhatsApp message three days before the attack.
Butt did not fall into that category when they last investigated him.
One of the messages behind the training is "run, hide, tell", in case of an attack. Borough Market, a popular gathering place, remains closed as more evidence is gathered.
"Some of the people who were here on Saturday night are still a bit fragile", said Leon chief executive John Vincent.
However, some nearby businesses have started to reopen.
Thomas, 45, was walking with his girlfriend over the bridge when the attack began on Saturday night. "Our heartfelt thoughts and prayers are with the victims and families", they said.
CNN's Sandi Sidhu met him several times while reporting on the group in the United Kingdom between 2014 and 2016.
The mother of the Italian-Moroccan jihadi who helped carry out a deadly attack in London last week said on Tuesday she had tried to keep her son from falling under the sway of Islamic State ideology, but the internet and his London friends changed him.
Spanish authorities have said Ignacio Echeverria, a 39-year-old banking analyst who fought off terrorists with his skateboard, was also murdered. Amato told Italy's Radio 24 that Zaghba was flagged to British authorities as a "possible suspect". The next of kin of 45-year-old Xavier Thomas were informed. He told the BBC: "People are going to look at the front pages today and they're going to say, 'How on earth could we have let this guy or possibly more through the net?'"
London Mayor Sadiq Khan said questions would need to be asked about what the police knew about Butt.
The British Transport Police officer, whose name was not disclosed, was stabbed in the head, face and leg and is still hospitalized.
She said she has been "humbled and privileged" to hear of the heroic actions of members of the public and the emergency services in their response to the attack. Although he did not say anything particular, I felt it in his voice.
Over the past two years, Italy has expelled 181 people who were suspected of extremist activities but for whom there was insufficient evidence to bring formal charges.
The footage believed to be from a surveillance camera emerged on social media.
The video shows a man crossing a deserted street in the market.
The identities of the two attackers announced Monday fit a pattern. The man was held on suspicion of preparing terrorist acts and was taken into custody.
Mr Pigeard, born in Paris, was one of three French victims among the eight people killed in London Bridge.
Law enforcement have conducted a series of raids in connection with the London Bridge attack and arrested 19 people in the week after.