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I understand that the Random variable is a function such as, $RV:\sigma-field(\Bbb C) \rightarrow range \;of \;random\;variable \in\Bbb R$ where $\Bbb C$ denotes sample space.

Is there any particular name for the range of random variable?

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The set of possible values that random variable can take is called its support. I don't think that there is any name for range. Support is a broader and more precise term. For example, say that random variable can take only non-negative, integer values -- range wouldn't tell us that we are interested only in integers. Cantor distribution is an example of distribution with complicated support.

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