Zapiski Nauchn. Semin. POMI,

English translation: J. Math. Sci. (N.Y.),

**Abstract.**
We present a sketch of the Fourier theory on the infinite symmetric group
$S_\infty$. As a dual space to $S_\infty$, we suggest the space (groupoid) of Young bitableaux B. The Fourier transform of a function on the infinite symmetric group is a martingale with respect to the so-called full Plancherel measure on the groupoid of bitableaux. The Plancherel formula determines an
isometry of the space $l^2(S_\infty,m)$ of square summable functions on the infinite symmetric group with the counting measure and the space
$L^2(B,\tilde \mu)$ of square integrable functions on the groupoid of bitableaux with the full Plancherel measure.

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