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How is $I_{[0,\infty)}(t)$ defined? This must be a notation in probabilty theory.

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It's likely an indicator function: it has value 1 on $[0,\infty)$ and 0 on $(-\infty,0)$. –  David Mitra Jul 4 '12 at 12:55
    
Although indicator functions come up in probability, it doesn't seem that an indicator over an infinite interval would come up. –  Michael Chernick Jul 4 '12 at 15:26
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You can use the Heaviside Step Function $H(t)$:

$H(0) = 1$ is used when $H$ needs to be right-continuous. For instance cumulative distribution functions are usually taken to be right continuous, as are functions integrated against in Lebesgue–Stieltjes integration. In this case $H$ is the indicator function of a closed semi-infinite interval: $$ H(t) = \mathbf{I}_{[0,\infty)}(t).\, $$

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...or one could use Iverson brackets as well: $[t \geq 0]$. –  J. M. Aug 9 '12 at 10:49
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