The incident happened near to Wamphray Farm on North Berwick Law at around 5pm on Sunday, 22 July.
A disused building containing a number of household appliances was deliberately set on fire causing considerable damage to the building and surrounding area.
The Scottish Fire and Rescue Service extinguished the blaze and no one was injured as a result of the incident.
Officers are now urging anyone who can assist with inquiries to come forward.
Inspector Andrew Harborow said: “This deliberate act has caused damage to the owner’s belongings and property.
“Thankfully no one was injured, however, deliberately starting fires can have catastrophic consequences.
“I would remind the public that disused buildings should not be used as an area to congregate or enter without the owner’s permission.
“Anyone who may have seen any suspicious behaviour in the North Berwick Law area on Sunday evening, or has any information to assist with our investigations, is asked to get in touch with officers as soon as possible.”
Those with information can contact Police Scotland via 101, quoting incident number 2990 of 22 July, or report this anonymously to Crimestoppers on 0800 555 111.
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