Peggy Lee had the quintessential unhappy childhood in small-town North Dakota: dead mother (when Lee, then Norma Deloris Egstrom, was 4), alcoholic father, wicked stepmother (who started out as the stern housekeeper) and two devastating house fires in six years. (Lee didn’t write the lyric “Is that all there is to a fire?” but every time she sang it, she certainly knew the answer.)

James Gavin’s eminently readable biography, “Is That All There Is?: The Strange Life of Peggy Lee,” has one aspect of decided strangeness itself. The book could have been called “Peggy Lee Was a Big, Fat Liar.” If the scores of people Gavin interviewed are to be believed, Lee, who died in 2002 at 81, invented health problems (before she had real ones), claimed credit for songs she didn’t write, built lawsuits on outrageously fabricated events and made up two of the major elements of her mythology: that Lee’s stepmother beat her and that Lee and her first husband had decided to remarry right before he died.

Still, there’s something admiring in this account of a grossly unhappy woman’s persistence, of carefully nurtured and staged talent, of her “Lady and the Tramp” lawsuit against Disney. In ­summary: Miss Piggy, the temperamental Muppet, began life as Miss Piggy Lee (Lee threatened to sue). But there is an understanding that Lee was not the first celebrity to practice serial egocentric cruelty.

Gavin, the author of the Lena Horne biography “Stormy Weather,” can tell a good story, so even his early chapters (often the deadliest part of celebrity biographies) are engrossing. The stepmother probably did detest Lee and did abuse her emotionally. The story picks up steam at the same time Lee’s singing career does. Benny Goodman hired her as the vocalist for his big-band-era orchestra in 1941. Lee had hits like “Why Don’t You Do Right?,” “It’s a Good Day,” “Lover” and “Fever,” her biggest prime-of-life success, which was considered shockingly sexual. She tried movies, starring in a 1952 remake of “The Jazz Singer” and earning a 1955 Oscar nomination for playing a singer with mental problems in “Pete Kelly’s Blues.” But singing and songwriting were her gifts, and she spent decades as the adored and highly paid queen of intimate nightclubs — while fighting her own irrelevance when rock ’n’ roll and disco came along. The baby boomer generation remembers her for “Is That All There Is?,” Lieber and Stoller’s ballad of existential angst that amazed almost everyone by becoming a major hit in 1969.