How to remove all characters after first space for OLAP dimension members

You should use INSTR function (instead of CHARINDEX), so something like this on Adventure Works database:

with member Measures.[Short Name]
as
left("Aaron A. Allen", instr("Aaron A. Allen", " " ) - 1)

select Measures.[Short Name] on 0,
[Customer].[Customer].members on 1
from [Adventure Works];

or like this:

with member Measures.[Customer Short Name]
as
left([Customer].[Customer].currentmember.MEMBER_CAPTION, instr([Customer].[Customer].currentmember.MEMBER_CAPTION, " " ) - 1)

select {Measures.[Customer Short Name], [Measures].[Customer Count]} on 0,
[Customer].[Customer].members on 1
from [Adventure Works]