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Infinitesimal Calculus of Random Walk and Poisson Processes

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Gauge Institute Journal,<br />

H. Vic Dannon<br />

Consequently, there are countably many real numbers in the<br />

interval<br />

[,] ab<br />

, <strong>and</strong> the Integration Sum has countably many<br />

terms.<br />

While we do not sequence the real numbers in the interval,<br />

the summation takes place over countably many f ( xdx. )<br />

The Lower Integral is the Integration Sum where f ( x ) is<br />

replaced<br />

by its lowest value on each interval<br />

3.2<br />

∑<br />

x∈[ a, b]<br />

⎛<br />

⎜⎝<br />

dx dx<br />

2 2<br />

[ x − , x + ]<br />

⎞<br />

inf f ( t)<br />

dx<br />

⎠⎟<br />

x− ≤t≤ x+<br />

dx dx<br />

2 2<br />

The Upper Integral is the Integration Sum where f ( x ) is<br />

replaced by its largest value on each interval<br />

3.3<br />

∑<br />

x∈[ a, b]<br />

⎛<br />

⎜⎝<br />

⎞ sup f ( t)<br />

dx<br />

⎠⎟<br />

dx dx<br />

2 2<br />

x− ≤t≤ x+<br />

[ x − , x + ]<br />

dx dx<br />

2 2<br />

If the integral is a finite hyper-real, we have<br />

14

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