Lars von Trier's 2000 film Dancer in the Dark is the story of Selma Jezkova, a Czech immigrant who works in a factory in the state of Washington circa 1964. Selma suffers from severe myopia, to the point that she will soon be totally blind. Selma secretly hoards cash from her job in order to pay for an operation for her son Gene who has the same affliction.
Selma is a perpetually innocent person, loyal to her dying breath, despite others not being loyal to her. She doesn't even testify on her own behalf in court during her own murder trial, lest that break her oath to Bill, the murder victim. She works in a dead end job for little pay, occasionally taking on double shifts to earn more money for her son. She doesn't argue or make a fuss and is always polite, to a fault. This is in contrast to Laura Mulvey's assertion that female protagonists always behave like men. In other words, Selma is pretty naive.
The film is structured as a dark homage to Hollywood musicals. Selma's sunny personality could have been lifted right from a 1940s song and dance film like Ziegfeld Follies or a Busby Berkeley movie. Throughout the film, the drama is interrupted with musical numbers that take place in Selma's imagination. However, von Trier turns the old Hollywood optimism on its head with Dancer's realistic depiction of poverty, hardship and injustice.
Karl Marx's theory of alienation fits into certain aspects of the plot of Dancer in the Dark. Marx believed that capitalism deliberately alienated people from one another, particularly laborers, in order to take advantage of them (Reader in Marxist Philosophy p. 297). Human nature is manipulated to turn peers against each other for the sake of profit.
First, what is the significance of this alienation in the film? First, due to her daydreaming, naiveté and near blindness, Selma is already alienated from much of society. Her employer takes advantage of her (and its other employees) by the hard labor and low pay. Bill, the police officer killed by Selma, represents the government or bourgeois class in society (those with the power). Bill is in a bad spot with his wife and house because he is behind on payments to the bank, which serves as another indictment to capitalism and the status quo. Bill takes advantage of his police position by bullying Selma and taking her money to solve his own problems. She "defeats" this power, but reluctantly and at her own peril. The rest of the law establishment responds by arresting, imprisoning, trying, and convicting her. Her friends from the proletariat want to help her but their own ability is limited. Ultimately she is hanged and justice is not served.
Secondly, what is the significance of Selma's blindness? I think this establishes two things: why she is so innocent and why she is so in love with musicals. She cannot clearly see all the problems and dreariness around her, so she retains a friendly, charming demeanor, perhaps even childish. It also explains why she enjoys her daydreams so much. She may not be able to enjoy standard Hollywood movies, but musicals have so much movement and audio stimulation that she can still get some enjoyment from it. With her eyesight more or less cut off, her mind can take over and take her into her own musical numbers.
Lastly, what is the significance of those musical numbers interspersed among the dramatic scenes? These serve a few purposes. They are levity to break up the heavy, sad scenes of oppression in Selma's life. They also provide a window into Selma's imagination. She has a genuine personality to the audience. The scenes show that Selma is the most human of the characters because of her unwillingness to feel sorry for herself or give up. During her tedious, rough job at the factory she can escape temporarily into a dance routine. During the musical sequences, the film (actually digital video tape) is brighter and more colorful, and not so drab as in the "real life" scenes. Therefore the songs are more real to the audience than the main plot. The film seems to be saying that life may be hard, but your attitude matters too. It isn't quite so hard if you're creative and have something fun to keep you company.