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*The pointwise maximum of two subharmonic functions is subharmonic.
*The limit of a decreasing sequence of subharmonic functions is subharmonic (or identically equal to <math>-\infty</math>).
*Subharmonic functions are not necessarily continuous in the usual topology, however one can introduce the [[fine topology]] which makes them continuous.
==Examples==
If <math>f</math> is [[analytic]] then <math>\log|f|</math> is subharmonic. More examples can be constructed by using the properties listed above,
by taking maxima, convex combinations and limits. In dimension 1, all subharmonic functions can be obtained in this way.
==Riesz Representation Theorem==
If <math>u</math is subharmonic in a region <math>D</math>, in [[Euclidean space]] of dimension <math>n</math>, <math>v</math> is harmonic in <math>D</math>, and <math>u\leq v</math>, then <math>v</math>
is called a harmonic majorant of <math>u</math>. If a harmonic majorant exists, then there exists the least harmonic majorant, and
:<math>u(x)=v(x)-\int_D\frac{d\mu(y)}{|x-y|^{n-2},\quad n\geq 3</math>
while in dimension 2,
:<math>u(x)=v(x)+\int_D\log|x-y|d\mu(y).</math>
This is called the [[Friedrich Riesz|Riesz]] representation theorem.
==Subharmonic functions in the complex plane==
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