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Murray said on Saturday that he will play in Shenzhen and Beijing, but the former world number one will miss the final two ATP events in Shanghai and Paris.
While the women's rules already call for a break if the conditions are too extreme, there is no such provision for the men.
There was plenty of criticism of Rask's ruling on social media, including from three-time major champion Andy Murray's mother, Judy.
Andy Murray, the 2012 US Open winner, who was playing his first Grand Slam since Wimbledon of last year, is still fighting to find full fitness after hip surgery in January.
Andy Murray, the 2012 US Open champion is playing just his fifth tournament of the year after undergoing hip surgery in January.
For the first time since Wimbledon in June 2017, a tournament will have the entire Big Four in the field: Roger Federer, Rafael Nadal, Novak Djokovic and Andy Murray.
Murray, who won the first of his three Grand Slam titles at Flushing Meadows in 2012, arrives this year ranked 378th in the world and still working his way back from hip surgery in January.
Andy Murray's injury comeback was sidetracked as the three-time Grand Slam champion lost 6-1, 1-6, 6-4 to Lucas Pouille.
Novak Djokovic and Venus Williams also were among the Grand Slam champions who missed out on a first-round bye at the last major tuneup event for the US Open, which starts August 27 in New York.
The former world number one is now ranked 375 as he continues his comeback from an 11-month injury layoff.