On the cardinal number of the family of all invariant extensions of a nonzero σσ-finite invariant measure

It is shown that, for any nonzero σσ-finite translation invariant (translation quasi-invariant) measure μμ on the real line R, the cardinality of the family of all translation invariant (translation quasi-invariant) measures on R extending μμ is greater than or equal to 2ω12ω1, where ω1ω1 denotes the first uncountable cardinal number. Some related results are also considered.