It is interesting to see Dexter believing Judy as "his own" because it reflects Fitzgerald's assertion that some dreams are powerful enough to control the beholder as opposed to the individual controlling it. What this ends up meaning is that Dexter truly believes that he controls his ability to dream and revere Judy and what she represents to him. In reality, he is being controlled by the pursuit of an exterior end that has little, if anything, to do with him. He is trapped by the allure of wealth, privilege, and established money and seeks to enter this world. In this process of coveting, the reader understands that Dexter is not in control of it, as it is in control of him.
Dexter’s thinking of Judy as “his own” is completely ironic because throughout most of the story, she has no real regard for him. When they first meet as children, Judy simply wants Dexter to be her caddy and continues to call him “Boy” even though he is clearly older than she is. Judy never even bothers to ask Dexter his name. When they meet again later in life, Judy uses Dexter and his attentions for selfish motives. He is quite smitten by her, but Judy treats Dexter with wavering indifference. Judy is a reckless character, one who could never really “belong” to anyone but herself.