Murray beats Isner 6-3 6-7 (4-7) 6-4
Victory gives Scot his seventh title of 2016
He will become new world number one on Monday
Isner 3-6 2-2 Murray*
Murray, in a navy blue t-shirt, white shorts and white cap, forces opponent Isner to go long with a return which is frustrating for the American, who angrily thrashes the ball back.
The Scot then goes 40-15 up in the game with a return that flicks the top of the net, with the pace coming off the ball, and apologises to Isner for that bit of fortune, before levelling the set.
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*Isner 3-6 2-1 Murray
Murray finished the last game with an ace, Isner hits back with one of his own in this.
The American has an easy drop shot from close in, but the crowd cannot believe it as he hits the net. There was an open court for him to pop the ball in to.
It does not do too much damage though, as he takes the game and heads back to his chair.
Isner 3-6 1-1 Murray*
Murray serving for the first time in the second set as Isner strikes the net and hits a couple too long as the Scot goes 40-0 up.
An ace to the left of Isner gives him the game in quick fashion.
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Thomas Alway: Congratulations Andy Murray - in my eyes the best British sportsman of all time
Laura Franchi: Arise Sir Andy Murray, let's face it you deserve it. Incredibly proud to see what you have achieved
Andrew Johnson: His form this year has been immense. He has to be a certainty for SPOTY now surely!!
*Isner 3-6 1-0 Murray
Isner opens up with an ace and there's a bit of back and forth before Murray shovels a half-volley into the net, leading to a shake of the head.
The American looks more confident coming up to the net, and beats Murray twice with two forehands to hold his serve.
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Isner 3-6 *Murray
Murray to serve for the first set and a stunning backhand to a cracker from Isner has the crowd on their feet.
Isner draws it back to 15-15 with an enormous forehand before he skews his forehand and sends the ball into the crowd.
A shake of the head from Murray as he hits the net but an ace gives him a set point, despite Isner's challenge.
Murray responds with a stinging serve that catches the top of Isner's racquet and spins out of court. First set to Murray.
Image source, Getty Images*Isner 3-5 Murray
Isner releases some anger with an ace, before Murray, on one knee, slices his return into the net.
Isner whizzes through the game to hold his serve.
Isner 2-5 *Murray
Murray opens with a double fault and then can't get near a thudding forehand from Isner, and he slams his racquet into his trainer in frustration.
Isner steps it up further with another winning forehand to go up two break points, but Murray claws back with a cheeky volley over the net.
Isner can't believe it, and he's even less impressed when he slams a return into the net for deuce.
Murray serves a fast, furious ace that Isner can't get to, and the American forces a backhand into the net and Murray, somehow, holds.
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Emma Scott: Finally sitting down and freezing cold on a really delayed train back to London, but at least I have #bbctennis updates. #GoAndy
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Russell Fuller
BBC tennis correspondent on BBC Radio 5 live sports extra
Murray is seeking his fourth title in five weeks, which is particularly remarkable after the exertions of his fantastic summer. Victory here would follow wins in Beijing, Shanghai and Vienna.
*Isner 2-4 Murray
Isner unfurls a backhand that Murray sends long before another big serve takes him to 30-0.
The American slices Murray's forehand wide then misses another volley as he and Murray get up to the net again.
That's knocked Isner. A double fault hands Murray a break point and Isner misjudges a backhand. That's long, that's out, and Murray's broken him.
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Matt Craig: Well done Andy Murray - world number 1! A fantastic champion but now officially the best in the world
Jonathan Billing: Congrats Andy Murray. Thoroughly deserved after all your hard work
Adam Sofroniou: Murray has to be recognised as the greatest British sportsman in recent times. One of the very best ever
Isner 2-3 *Murray
Plenty of back and forth as Isner forces an error out of Murray, Murray responds with a stinging ace and then the American takes the lead with a clever crosshand.
Murray draws it back to 30-30 before Isner misjudges a shot at the net and hits it with a volley! What an opportunity.
In fact, Murray follows suit, slashing a forehand into the net for deuce.
Murray takes the first rally before the forehand proves a clean winner.
Russell Fuller
BBC tennis correspondent on BBC Radio 5 live sports extra
The first two drop shot exchanges have gone John Isner's way, that is not what you would expect. He made very good ground to get to them.
Image source, Getty Images*Isner 2-2 Murray
Isner sends Murray the wrong way with a serve volley, which leaves the Brit swishing at the ground in annoyance.
A lovely drop shot puts Isner up but Murray returns the serve with a cross-court that surprises him and he steers it into the net.
Murray can't keep Isner's serve out for long though, and the American holds.
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Isner 1-2 *Murray
Murray cramps Isner up for room before the players come up to the net for a brief rally.
Murray is quick on his feet and outmanoeuvres the American, before forcing Isner to backhand into the net as he holds for love.
*Isner 1-1 Murray
Isner has served more aces than anybody in this tournament and he beats Murray with a sizzler.
He rounds things off with his third ace to hold to 15.