Extensions of Haar measure to relatively large nonmeasurable subgroups

Abstract:

Let G be a locally compact group, with $\lambda$ a left Haar measure on G. A subgroup is large if it has finite index; a relatively large subgroup of G is a large subgroup of an open subgroup. In §1 we have an existence theorem for relatively large nonopen subgroups, and we observe that such subgroups are not $\lambda$-measurable. This motivates the development, in §2, of a left-invariant countably additive extension ${\lambda ^ + }$ of $\lambda$, which includes in its domain all unions of left translates of a given relatively large subgroup K. For an arbitrarily chosen family ${\mathcal {K}_I}$ of relatively large subgroups of G, we define (in §3) a finitely additive measure $\lambda _I^ +$ such that, for any $K \in {\mathcal {K}_I},\lambda _I^ +$ is an extension of the corresponding ${\lambda ^ + }$ defined in §2. An example shows that $\lambda _I^ +$ need not be countably additive. Finally, in §4, we observe some aspects of the relationship between ${\lambda ^ + }$-measurable and $\lambda$-measurable functions, in the context of existing literature on extensions of Haar measure. In particular, we generalize the well-known proposition that $\lambda$-measurable characters are continuous.