As he takes young Molly around the park near his London home, Murray has much to ponder. In the space of the past two weeks he had beaten Rafael Nadal for the first time – and done it over two rain-sodden days – and he had reached his first grand slam final.

That he lost in straight sets to Roger Federer is no disgrace – Federer was on a mission to win his first major title of the year. With the level of tennis he brought to the match, it is hard to imagine that anyone could have beaten him on Monday night.

When Murray came to New York, he stated that he had come here to win. As he leaves, three weeks on, he knows that he can win here in the future. In just 21 days and seven matches he has matured from being a young hopeful to a true contender.

"I was effectively three sets away from winning a Grand Slam," he said, "which is much closer than I was at this time last year, and I've worked very hard to get to this stage. Winning a slam is really the only goal I have left for myself. I'm going to have work harder, improve my game and get physically stronger if I'm going to do that.

"To play in front of a 23,000 crowd like that – it was packed out there – against Federer and be three sets away from winning a US Open is a good effort at my age, but I don't want to be remembered for losing in the final of the US Open.

"I want to win one of these tournaments and I'm going to try my hardest to do it."

He has been on a very steep learning curve this summer. The months of graft and grind in training have made him fitter and stronger than ever before, while every match at the US Open was another milestone and he has soaked up the experience like a sponge.

"Beating Nadal was probably the biggest achievement for me in such a big match and with what happened with the rain and being down in the fourth set as well. I just think that was mentally huge for me.

"I got killed by (Novak] Djokovic quite a few times and it's only been recently that I've understood how to play against him and figured out how to beat the top guys. This year I've beaten Federer, Djokovic, Nadal, Davydenko, Wawrinka a few times and those guys are all in the top 10. I guess I just need to do it in the big tournaments now and this was the first time I've done it."

New York is his favourite city and yet he has lived quietly here. Keeping away from his laptop and only taking the occasional phone call and text message, he has not allowed himself to become involved in the excitement surrounding his success. The fact that he has to be "pretty boring" during his grand slam challenges is another sign of his growing maturity.

That he was able to rattle Federer in the second set of the final and had the fight to break the champion as he served for the match is also a sign to his rivals that Murray is ready to win his first major title.

"I always thought he was a big match player," Federer said. "Give him the biggest court in the world, he will have no problem handling it. That's what he's shown anyway throughout his career."

The next grand slam is the Australian Open in January. Another hard court tournament, it is another chance for Murray to achieve his goal. With what he has learned here, he knows that he is a contender for the honours in Melbourne, Wimbledon and Flushing Meadows next year.

But for now he is just glad to be going home. "I'll go back and see my – or rather me and my girlfriend's – dog," he said. "I've been away for three weeks now, so I'll go and walk the dog."

Murray and Molly have a lot to think about as they stroll through the park and plan for the future.

Federer, meanwhile, has sent an ominous warning to his rivals by saying that Monday night's victory had put him at ease for a resurgence in 2009.

The 27-year-old Swiss lost his world No 1 ranking to Nadal in August after 12 defeats in 2008 which included the loss of his Australian Open and Wimbledon titles and another French Open final setback to the Spaniard. But he responded brilliantly in New York.

"I would have been disappointed losing and having three finals and one semis of the slams," Federer admitted.

"You feel like you missed an entire year, you know, being so close but yet so far, because semis and finals don't help me a whole lot any more in my career.

"It's all about the wins, and that's why this is huge. This is massive, really, and I'm very, very happy about this grand slam obviously. It's a different type of flavour, this one, to me, no doubt. And I can definitely go into the rest of the season more relaxed now, and then also looking forward with great spirits for next year."