A function f(x) has the value of zero expect one point, where the value is infinite. Does the integral of f(x) equal 0? Or any other values? Thanks a lot.
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What sort of integral are you using? – copper.hat Nov 11 '13 at 23:30
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I am not sure what sort it is.... I am calculating the expectation of a uniform random variable, the variable has a value of infinity at one point while keeps zero at other parts. – monkinsane Nov 11 '13 at 23:39
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Most likely the Lebesgue integral then, see the answers below. – copper.hat Nov 11 '13 at 23:40
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It sounds like you are thinking about the Dirac delta "function" which is not a proper function. Changing the value of a function at one point cannot change its integral. It is not allowed in $\Bbb R$ to have the value of a function be infinite. You can explore this term on this site and elsewhere to learn its properties.
Ross Millikan
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Assuming that you're talking about Lebesgue integration (since we cannot take the Riemann integral of an unbounded function), then the integral is $0$. This is because the function is equal to the zero function except at a single point, and a singleton is a set of measure $0$, so we ignore what happens there.
Now, for example, the Dirac delta function can be loosely thought of as a function that is zero everywhere on the real line, except at $0,$ where it is infinite, and has an integral over the whole real line of $1,$ but again, this is just a rough idea. It is more accurately defined as a measure/probability distribution.
Cameron Buie
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So you mean following the example of Dirac delta function, my integral is not zero? Thank you! – monkinsane Nov 11 '13 at 23:47
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If you're dealing with a uniform distribution whose density is taken to be $\infty$ at one point and $0$ in all the others, then you have what is often referred to as a "point mass distribution," and (as a probability distribution) has a total integral of $1.$ This is the same sort of "loose idea" that is behind the Dirac delta. – Cameron Buie Nov 11 '13 at 23:56
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The problem is like this: I am calculating the expectation of f(x) on interval[0,1], and x has uniform distribution on interval[0,1], so pdf of x is 1 for each point in this interval. f(x) is infinite at point 0, while it is 0 at (0,1]. – monkinsane Nov 12 '13 at 00:10
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Then $f$ is exactly the Dirac delta. Effectively, it is an indicator function of sorts, that lets us know whether or not $0$ is in a given set. For any measurable subset $A$ of the real line, we have $$\int_Af(x),dx=\begin{cases}1 & \text{if }0\in A\0 & \text{if }0\notin A.\end{cases}$$ – Cameron Buie Nov 12 '13 at 00:19
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