# Kemperman's theorem

## Contents

## Statement

### For two subsets

Suppose is a compact connected topological group with a Haar measure of total volume 1 (so that the Haar measure is a probability measure). Suppose are compact subsets of . Then, the product of subsets is also a compact subset and:

### For finitely many subsets

Suppose is a compact connected topological group with a Haar measure of total volume 1 (so that the Haar measure is a probability measure). Suppose are compact subsets of . Then, the product of subsets is also a compact subset and:

### Note for abelian groups

When, satisfies the additional condition of being an abelian group, and the group operations are denoted additively, the product of subsets is termed the Minkowski sum. Thus, this provides a lower bound on the size of the Minkowski sum.

## Caveats

### Importance of connectedness

Connectedness is important because if has a proper open subgroup, then by compactness it must have finite index, and then the measure of the subgroup is the reciprocal of that index. Taking to both be that subgroup violates the theorem.

### Sharpness of estimate

We can take to be the circle group and to be arcs starting at the identity element to show that the estimates cannot be improved.

## Related facts

- Product of subsets whose total size exceeds size of group equals whole group
- Cauchy-Davenport theorem: A similar fact for a group of prime order.