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Posts Tagged ‘Measure Theory’

Areas of Mathematics

October 22, 2013 3 comments

For one of my upcoming talks I am trying to include an exhaustive mindmap showing the different areas of Mathematics, and somehow, how they relate to each other. Most of the information I am using has been processed from years of exposure in the field, and a bit of help from Wikipedia.

But I am not entirely happy with what I see: my lack of training in the area of Combinatorics results in a rather dry treatment of that part of the mindmap, for example. I am afraid that the same could be told about other parts of the diagram. Any help from the reader to clarify and polish this information will be very much appreciated.

And as a bonus, I included a \LaTeX script to generate the diagram with the aid of the tikz libraries.

\tikzstyle{level 2 concept}+=[sibling angle=40]
\begin{tikzpicture}[scale=0.49, transform shape]
  \path[mindmap,concept color=black,text=white]
    node[concept] {Pure Mathematics} [clockwise from=45]
      child[concept color=DeepSkyBlue4]{
        node[concept] {Analysis} [clockwise from=180]
          child { 
            node[concept] {Multivariate \& Vector Calculus}
              [clockwise from=120]
              child {node[concept] {ODEs}}}
              child { node[concept] {Functional Analysis}}
              child { node[concept] {Measure Theory}}
              child { node[concept] {Calculus of Variations}}
              child { node[concept] {Harmonic Analysis}}
              child { node[concept] {Complex Analysis}}
              child { node[concept] {Stochastic Analysis}}
              child { node[concept] {Geometric Analysis}
                [clockwise from=-40]
                child {node[concept] {PDEs}}}}
          child[concept color=black!50!green, grow=-40]{ 
            node[concept] {Combinatorics} [clockwise from=10]
              child {node[concept] {Enumerative}}
              child {node[concept] {Extremal}}
              child {node[concept] {Graph Theory}}}
          child[concept color=black!25!red, grow=-90]{ 
            node[concept] {Geometry} [clockwise from=-30]
              child {node[concept] {Convex Geometry}}
              child {node[concept] {Differential Geometry}}
              child {node[concept] {Manifolds}}
              child {node[concept,color=black!50!green!50!red,text=white] {Discrete Geometry}}
              child {
                node[concept] {Topology} [clockwise from=-150]
                  child {node [concept,color=black!25!red!50!brown,text=white]
                    {Algebraic Topology}}}}
          child[concept color=brown,grow=140]{ 
            node[concept] {Algebra} [counterclockwise from=70]
              child {node[concept] {Elementary}}
              child {node[concept] {Number Theory}}
              child {node[concept] {Abstract} [clockwise from=180]
                child {node[concept,color=red!25!brown,text=white] {Algebraic Geometry}}}
              child {node[concept] {Linear}}}
    node[extra concept,concept color=black] at (200:5) {Applied Mathematics} 
      child[grow=145,concept color=black!50!yellow] {
        node[concept] {Probability} [clockwise from=180]
          child {node[concept] {Stochastic Processes}}}
      child[grow=175,concept color=black!50!yellow] {node[concept] {Statistics}}
      child[grow=205,concept color=black!50!yellow] {node[concept] {Numerical Analysis}}
      child[grow=235,concept color=black!50!yellow] {node[concept] {Symbolic Computation}};
\end{tikzpicture}
Categories: Algebra, Analysis, Applied Mathematics, Combinatorics, differential equations, Geometry, Linear Algebra, Probability, Scientific Computing, Statistics, TeX/LaTeX Tags: , , , , , , , , , , , , , , , ,

A nice application of Fatou’s Lemma

June 1, 2013 Leave a comment

Let me show you an exciting technique to prove some convergence statements using exclusively functional inequalities and Fatou’s Lemma. The following are two classic problems solved this way. Enjoy!

Exercise 1 Let {(X, \mathcal{F}, \mu)} be a measurable space and suppose {\{f_n\}_{n\in \mathbb{N}}} is a sequence of measurable functions in {L_1(\mu)} that converge almost everywhere to a function {f \in L_1(\mu)}, and such that the sequence of norms {\big\{ \lVert f_n \rVert_1 \big\}_{n \in \mathbb{N}}} converges to {\lVert f \rVert_1}. Prove that the sequence of integrals {\int_E \lvert f_n \rvert\, d\mu} converges to the integral {\int_E \lvert f \rvert\, d\mu} for every measurable set {E}.

Proof: Note first that

\displaystyle \begin{array}{rcl} \lvert f_n - f\rvert\, \boldsymbol{1}_E \leq \lvert f_n - f \rvert \leq \lvert f_n \rvert + \lvert f \rvert. \end{array}

Set then {g_n = \lvert f_n \rvert + \lvert f \rvert - \lvert f_n-f \rvert\, \boldsymbol{1}_E} (which are non-negative functions) and apply Fatou’s Lemma to that sequence. We have then

\displaystyle \begin{array}{c} \int_X \liminf_n g_n\, d\mu \leq \liminf_n \int_X g_n\, d\mu \\ \\ \int_X \liminf_n \big( \lvert f_n \rvert + \lvert f \rvert - \lvert f_n - f \rvert\, \boldsymbol{1}_E \big)\, d\mu \leq \liminf_n \int_X \big( \lvert f_n \rvert + \lvert f \rvert - \lvert f_n - f \rvert\, \boldsymbol{1}_E \big)\, d\mu \\ \\ \int_X \big( 2\lvert f \rvert - \limsup_n \lvert f_n-f \rvert\, \boldsymbol{1}_E\big)\, d\mu \leq \liminf_n \int_X \lvert f_n \rvert\, d\mu + \lVert f \rVert_1 - \limsup_n \int_E \lvert f_n -f \rvert\, d\mu \\ \\ 2\lVert f\rVert_1 \leq 2\lVert f \rVert_1 - \limsup_n \int_E \lvert f_n - f\rvert\, d\mu, \end{array}

which implies

\displaystyle \begin{array}{rcl} 0 \leq \liminf_n \int_E \lvert f_n-f \rvert\, d\mu \leq \limsup_n \int_E \lvert f_n-f \rvert\, d\mu \leq 0. \end{array}

It must then be {\int_E \lvert f_n - f \rvert\, d\mu = 0}. But this proves the statement, since

\displaystyle \begin{array}{c} \bigg\lvert \int_E \lvert f_n\rvert\, d\mu - \int_E \lvert f \rvert\, d\mu \bigg\rvert = \bigg\lvert \int_E \big( \lvert f_n \rvert - \lvert f \rvert \big)\, d\mu \bigg\rvert \\ \leq \int_E \Big\lvert \lvert f_n \rvert - \lvert f \rvert \Big\rvert\, d\mu \leq \int_E \lvert f_n - f \rvert\, d\mu \end{array}

\Box

Exercise 2 Let {(X, \mathcal{F}, \mu)} be a finite measure space and let {1<p<\infty}. Suppose that {\{ f_n \}_{n \in \mathbb{N}}} is a sequence of measurable functions in {L_p(\mu)} whose norms are uniformly bounded in {n} and which converge almost everywhere to a function {f}. Prove that the sequence {\big\{ \int_X f_ng\, d\mu \big\}_{n \in \mathbb{N}}} converges to {\int_x fg\, d\mu} for all {g \in L_q(\mu)} where {q} is the conjugate exponent of {p}.

Proof: The proof is very similar to the previous problem. We start by noticing that under the hypotheses of the problem,

\displaystyle \begin{array}{c} \bigg\lvert \int_x f_ng\, d\mu - \int_X fg\, d\mu \bigg\rvert = \bigg\lvert \int_X (f_n -f)g\, d\mu \bigg\rvert \\ \leq \lVert (f_n-f)g \rVert_1 \leq \lVert f_n - f \rVert_p\, \lVert g \rVert_q. \end{array}

If we prove that {\lim_n \lVert f_n-f \rVert_p = 0}, we are done.

We will achieve this by using the convexity of {\lvert \cdot \rvert^p}, since in that case it is

\displaystyle \begin{array}{rcl} \frac{\lvert f_n - f \rvert^p}{2^p} \leq \tfrac{1}{2} \lvert f_n \rvert^p + \tfrac{1}{2} \lvert f \rvert^p. \end{array}

hence,

\displaystyle \begin{array}{rcl} \lvert f_n -f \rvert^p \leq 2^{p-1} \big( \lvert f_n \rvert^p + \lvert f \rvert^p \big). \end{array}

Set then {g_n = 2^{p-1} \big( \lvert f_n\rvert^p + \lvert f \rvert^p\big) - \lvert f_n-f \rvert^p} (which are non-negative functions) and apply Fatou’s Lemma as before. \Box

Categories: Analysis, Teaching Tags: , ,