A bit of a newbie question but, caliber?

[Schoffleine]

What were the Lugers in WWII in terms of caliber? I found my dad's Luger the other day while house sitting and was trying to determine whether it was authentic or not (seems to be).

In my search, I found a site that said the caliber is printed on the underside of the barrel, and the one they were displaying was an 8.82. Well this one is an 8.84, evidently. Is that close enough to a 9mm to make no difference? I'm not sure I really want to fire this thing (don't know what would happen to it), but I am a bit curious as to whether it could be fired.

From what I can tell (I'd love to share pictures with y'all but don't have a transfer cable for my camera), it's a 1934 (or 1937, I've seen conflicting dates) Mauser-Werke Luger K-date with the S/42 markings. He has a P-08 holster too but some idiot took a knife to part of it and sawed off a rectangle (probably where an emblem was) on the backside. Still, it's in great shape for the age.

Oh, right, the caliber. Sorry, bit excited about this find.