Because i do not understand from how the play ended.
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Yes, the inspector is a fake. The police force doesn't have an Inspector Goole, and we never find out who he really was, but his character does have a huge part in foreshadowing future twists and turns in the plot. Some believe that the Inspector came to pre-warn the family, but that is debatable. Others believe that his name name Goole is a play on words, ghoul, which would could possibly mean he was otherworldly. Nonetheless, he wasn't a real inspector, and his actions allowed the family to see themselves and step up to do the right thing.
An Inspector Calls