If memory serves a locally compact semigroup admits a left invariant Borel measure iff it has a minimal ideal which is a direct product of a group and a right zero semigroup.

The measure is supported on the minimal ideal.

Update:

The result I was trying to recall is from http://www.ams.org/journals/proc/1966-017-02/S0002-9939-1966-0188341-7/S0002-9939-1966-0188341-7.pdf.

A locally finite Borel measure $\mu$ on a locally compact inverse semigroup $S$ is left invariant if $\mu(s^{-1}B)=\mu(B)$ for all Borel sets $B$. It is proved in the above paper that if $S$ is locally compact and left translations are proper (i.e. $s^{-1}K$ is compact when $K$ is compact, then $S$ has a left invariant measure iff it has a unique minimal left ideal. This is the same as saying it has a minimal ideal which is a direct product of a group and a right zero semigroup. You can take as the measure the product of Haar measure on the semigroup with your favorite locally finite Borel measure on the right zero semigroup.

If memory serves a locally compact semigroup admits a left invariant Borel measure iff it has a minimal ideal which is a direct product of a group and a right zero semigroup.

The measure is supported on the minimal ideal.

Update:

The result I was trying to recall is from http://www.ams.org/journals/proc/1966-017-02/S0002-9939-1966-0188341-7/S0002-9939-1966-0188341-7.pdf.

A locally finite Borel measure $\mu$ on a locally compact inverse semigroup $S$ is said to be left invariant if $\mu(s^{-1}B)=\mu(B)$ for all Borel sets $B$ where $s^{-1}B=\{x\in S\mid sx\in B\}$. 

It is proved in the above paper that if $S$ is locally compact and left translations are proper (i.e. $s^{-1}K$ is compact when $K$ is compact), then $S$ has a left invariant measure iff it has a unique minimal left ideal. This is the same as saying it has a minimal ideal which is a direct product of a group and a right zero semigroup. You can take as the measure the product of Haar measure on the semigroup with your favorite locally finite Borel measure on the right zero semigroup.

If memory serves a locally compact semigroup admits a left invariant Borel measure iff it has a minimal ideal which is a direct product of a group and a right zero semigroup.

The measure is supported on the minimal ideal.

If memory serves a locally compact semigroup admits a left invariant Borel measure iff it has a minimal ideal which is a direct product of a group and a right zero semigroup.

The measure is supported on the minimal ideal.

Update:

The result I was trying to recall is from http://www.ams.org/journals/proc/1966-017-02/S0002-9939-1966-0188341-7/S0002-9939-1966-0188341-7.pdf.

A locally finite Borel measure $\mu$ on a locally compact inverse semigroup $S$ is left invariant if $\mu(s^{-1}B)=\mu(B)$ for all Borel sets $B$. It is proved in the above paper that if $S$ is locally compact and left translations are proper (i.e. $s^{-1}K$ is compact when $K$ is compact, then $S$ has a left invariant measure iff it has a unique minimal left ideal. This is the same as saying it has a minimal ideal which is a direct product of a group and a right zero semigroup. You can take as the measure the product of Haar measure on the semigroup with your favorite locally finite Borel measure on the right zero semigroup.