In this paper we study functions with low influences on product probability spaces. The analysis of boolean functions f : {-1,1}^n \to : {-1,1} with low influences has become a central problem in discrete Fourier analysis. It is motivated by fundamental questions arising from the construction of probabilistically checkable proofs in theoretical computer science and from problems in the theory of social choice in economics.

We prove an invariance principle for multilinear polynomials with low influences and bounded degree; it shows that under mild conditions the distribution of such polynomials is essentially invariant for all product spaces. Ours is one of the very few known non-linear invariance principles. It has the advantage that its proof is simple and that the error bounds are explicit. We also show that the assumption of bounded degree can be eliminated if the polynomials are slightly "smoothed"; this extension is essential for our applications to "noise stability"-type problems.

In particular, as applications of the invariance principle we prove two conjectures: the "Majority Is Stablest" conjecture [29] from theoretical computer science, which was the original motivation for this work, and the "It Ain?t Over Till It?s Over" conjecture [27] from social choice theory. The "Majority Is Stablest" conjecture and its generalizations proven here, in conjunction with the "Unique Games Conjecture" and its variants, imply a number of (optimal) inapproximability results for graph problems.