Why would the Joker have half of the money?

He doesn't technically receive the money in a transaction sense.

His receipt of the money is 0 in actuality.

In the future should he kill Batman, he would receive 50%.

However at that point he had 100% of the money, one guy opposing him against his ever growing army of psychos, what do they care about money? What do they care if they rightfully received the money or not. The whole point of Joker getting "paid" to kill batman was just a ploy anyway to turn the mob upside down, the city upside down, and batman upside down anyway.

The Joker doesn't care about money. He doesn't care about fame. He doesn't care about life. He only cares about chaos.

Shortly after this act he states:

It's not about the money, its about sending a message

As his goons escort the mobster off to his demise. The Joker merely brought him there to show that he retrieved Lau and his money, only to shortly show the mobster how

All you care about is money. This city deserves a better class of criminal. And I'm gonna give it to them!

Joker doesn't want to kill Batman, he has too much fun with him, trying to show him how the natural order of the world is chaos.

You won't kill me out of some misplaced sense of self-righteousness, and I won't kill you because you're too much fun.

It can be argued that he knew of this fact long before he made his offer to the mobsters to kill Batman for 50% of their money. They were just another pawn in his long game for the soul of Gotham.

To clarify some points about the scene. All of the people there besides the mobster are under Joker's command. After escaping the police station with Lau, he most likely forced Lau to show him where he had stashed the money, which is most likely where they were to begin with. He then tied up Lau and contacted the mobster to retrieve his money from him. At that point the scene commences, Joker burns the money (and Lau) and has the mobster taken away to be fed to his own dogs.

At no point was the Joker actually "paid" for his services. He merely burns all of the money as a display of his wanton disregard for the value of currency.

Well if you go back to the original scene where the Joker met with the mob bosses for the first time, the conversations were the following :

It is simply we kill the batman

If it so simple why haven't you done it already

If you are good at something never do it for free

How much you want ?

Half

As far as I can understand he want half of their money to kill the batman not get half the money after killing batman.

Off course whether he really wants to kill batman or not is different question altogether.

The first time The Joker tries to make a deal, its for half the money to kill the Batman. Subsequently, we know that the Mob initially did not deal with the Joker (because they later decide to hire the clown). So we know that the Joker was working for the Mafia, although at this point we can be pretty much sure that they would certainly not give The Joker half of their money. It seems common sense to me.

Cut later, to the time The Joker is in the warehouse along with one of the Mafia boses. How did they get there? Joker obviously wriggled out the location out of Chao in the police holding cells. He then got out, went to the warehouse alongwith the Mafia boss. How did the Mafia boss got involved? Because the joker may have made a deal with him - he'll get all the money if they split it half-ways between them (The Joker always demands half!). It's not shown, but certainly implied. Why else would the Joker say my half? How else would the Mafia boss even be there if it was only the Joker who got the location out of Chao? From my interpretation, this seems to be the answer to your question.

I can't really source this answer, but my interpretation is that the Joker's words carry the implication that he is taking (and burning) his promised 50% cut in advance.

It's not an observation of fact that half the money is his or that the money in that pile was only half of all the pot.

It's a highly aggressive version of an old joke: "Can I have half that beverage","Sure", after chugging down the whole beverage "Sorry, my half was on bottom."

They were in fact saying, "I burned you money, but I burned mine too...see how unimportant it is to me? You can't buy my services - hahaha"

So a multi-layered joke I guess...sort of...

At least that's my interpretation.

Resurrecting this discussion somewhat, the question is fairly valid because it is strongly implied by the Chechen mobster that the Joker has in fact been paid. The Chechen asks the Joker what he is going to do with all his money, which clearly it wouldn't be if the Joker had not been paid as yet.

Also the theory that the Joker got half the money by promising to split all the money between himself and the Chechen falls down when the Joker asks "where's the Italian?" and therefore clearly refers to Sal Maroni and expects that Maroni would have been there too. In fact Maroni knew where the Joker was going to be and was likely invited to the warehouse by the Joker as Maroni offered to tell Gordon the location of the Joker while Gordon was visiting Harvey Dent in the hospital (he tells Gordon "you want him? I can tell you where he will be this afternoon").

So the question remains, why was Joker even assumed to have been given half of the money by the Chechen mob boss when Batman was quite clearly not dead?

The only reason I can come up with is that it was a major plot hole in the story OR that it was intended that upon being contacted by the Mob after Lau's capture that the Joker changed the terms and wanted half of the money in return for capturing Lau and returning their money to them. After all by then the Mob would be at risk of losing all of their money unless Lau was recaptured and as Gordon notes, the Joker intended to be caught in order to get into the same lock-up as Lau. No Lau = No Money for the Mob. So it would seem logical that a more desparate Mob would agree to even harsher terms by the Joker (he may well have promised them the return of Lau only for the money and that in the process Batman may get killed - if so he would have remained non-committal on killing Batman especially as by then he probably had no desire to kill Batman anymore).,

I suspect though that this was just a plot hole that may have been due to an oversight or to a backstory (like the one I postulated above) that wasn't properly explained in the film. Which is a pity because it would only have taken a few extra lines of dialogue to establish that there were new terms under which the Joker wanted half of the money.