Conditional Expectation, product of random variables












0












$begingroup$


I struggle with showing the following.



Let $tau$ be an exponentially distributed random variable, that means $mathbb P(tau > x)=mathrm{exp}(-x)$ and $xi$ a bounded random variable.



Show that $$mathbb Eleft[xifrac{mathbb 1_{(x<tauleq y)}}{tau}right]=0quad text{for all $x,y$ with $0<x<y$}$$
implies
$$mathbb Eleft[ximidtauright]=0.$$



Maybe you can help me with that.



Thank you very much!






random-variables conditional-expectation







share|cite|improve this question











$endgroup$

















    0












    $begingroup$


    I struggle with showing the following.



    Let $tau$ be an exponentially distributed random variable, that means $mathbb P(tau > x)=mathrm{exp}(-x)$ and $xi$ a bounded random variable.



    Show that $$mathbb Eleft[xifrac{mathbb 1_{(x<tauleq y)}}{tau}right]=0quad text{for all $x,y$ with $0<x<y$}$$
    implies
    $$mathbb Eleft[ximidtauright]=0.$$



    Maybe you can help me with that.



    Thank you very much!






    random-variables conditional-expectation







    share|cite|improve this question











    $endgroup$















      0












      0








      0


      1



      $begingroup$


      I struggle with showing the following.



      Let $tau$ be an exponentially distributed random variable, that means $mathbb P(tau > x)=mathrm{exp}(-x)$ and $xi$ a bounded random variable.



      Show that $$mathbb Eleft[xifrac{mathbb 1_{(x<tauleq y)}}{tau}right]=0quad text{for all $x,y$ with $0<x<y$}$$
      implies
      $$mathbb Eleft[ximidtauright]=0.$$



      Maybe you can help me with that.



      Thank you very much!






      random-variables conditional-expectation







      share|cite|improve this question











      $endgroup$




      I struggle with showing the following.



      Let $tau$ be an exponentially distributed random variable, that means $mathbb P(tau > x)=mathrm{exp}(-x)$ and $xi$ a bounded random variable.



      Show that $$mathbb Eleft[xifrac{mathbb 1_{(x<tauleq y)}}{tau}right]=0quad text{for all $x,y$ with $0<x<y$}$$
      implies
      $$mathbb Eleft[ximidtauright]=0.$$



      Maybe you can help me with that.



      Thank you very much!






      random-variables conditional-expectation




      random-variables conditional-expectation






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Dec 27 '18 at 10:07







      Agnetha Timara

















      asked Dec 27 '18 at 10:00









      Agnetha TimaraAgnetha Timara

      1087




      1087






















          1 Answer
          1






          active

          oldest

          votes


















          1












          $begingroup$

          The result is true for any positive random variable $tau$. Exponential distribution is note required. $E(xi |tau)=0$ iff $E(frac {xi} {tau} |tau)=0$ iff $E(frac {xi} {tau} I_A)=0$ for any set $A$ of the type ${tau in E}$ where $E$ is a Borel set in $mathbb R$. It is enough
          to prove this when $E$ is an open interval and in this case what we need is exactly the hypothesis.



          PS: there is no guarantee that $frac {xi} {tau}$ is integrable. To avoid this problem we can replace $xi$ by $xi I_{{tau >epsilon}}$ in the proof and let $epsilonto 0$ at the end.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Thank you very much for your quick reply. Isn't it necessary for $E(ximidtau)=0iff Eleft(fracxitaumidtauright)=0$ to hold that we have $Eleft|fracxitauright|<infty$ which does not have to be the case here.
            $endgroup$
            – Agnetha Timara
            Dec 27 '18 at 10:29










          • $begingroup$
            @AgnethaTimara You are right but that can easily be circumvented. It is enough to to show that $E( xi I_{tau >epsilon}|tau)=0$ for each $epsilon >0$.
            $endgroup$
            – Kavi Rama Murthy
            Dec 27 '18 at 10:37












          • $begingroup$
            Thanks again for clarifying that. Can you also tell me why it suffices to show that for a generating set of the $sigma$ algebra $sigma(tau)$. Is there a theorem I can refer to or can I use dominated convergence?
            $endgroup$
            – Agnetha Timara
            Dec 27 '18 at 11:44










          • $begingroup$
            @AgnethaTimara The $pi -lambda$ Theorem or Dynkin's Theorem is ideally suited for this kind of situation. It tells you immediately why it is enough to consider intervals.
            $endgroup$
            – Kavi Rama Murthy
            Dec 27 '18 at 11:47










          • $begingroup$
            Sorry for the many questions. I really appreciate your help. These theorems tell me that the intervals generate the Borel $sigma$ algebra. I don't understand yet why $int_{{x<tauleq y}} Z=0$ implies $int_{{tauin E}} Z=0$ for $Ein mathcal B$ and $Z$ a random variable. Obviously $E$ can be represented as a union / intersection of intervals but for example $int_A Z=0$ for $Ain{A_1,ldots A_n}$ does not imply $int_{bigcup_i A_i}Z=0$
            $endgroup$
            – Agnetha Timara
            Dec 27 '18 at 12:16












          Your Answer








          StackExchange.ready(function() {
          var channelOptions = {
          tags: "".split(" "),
          id: "69"
          };
          initTagRenderer("".split(" "), "".split(" "), channelOptions);

          StackExchange.using("externalEditor", function() {
          // Have to fire editor after snippets, if snippets enabled
          if (StackExchange.settings.snippets.snippetsEnabled) {
          StackExchange.using("snippets", function() {
          createEditor();
          });
          }
          else {
          createEditor();
          }
          });

          function createEditor() {
          StackExchange.prepareEditor({
          heartbeatType: 'answer',
          autoActivateHeartbeat: false,
          convertImagesToLinks: true,
          noModals: true,
          showLowRepImageUploadWarning: true,
          reputationToPostImages: 10,
          bindNavPrevention: true,
          postfix: "",
          imageUploader: {
          brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
          contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
          allowUrls: true
          },
          noCode: true, onDemand: true,
          discardSelector: ".discard-answer"
          ,immediatelyShowMarkdownHelp:true
          });


          }
          });














          draft saved

          draft discarded


















          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3053763%2fconditional-expectation-product-of-random-variables%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown

























          1 Answer
          1






          active

          oldest

          votes








          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          1












          $begingroup$

          The result is true for any positive random variable $tau$. Exponential distribution is note required. $E(xi |tau)=0$ iff $E(frac {xi} {tau} |tau)=0$ iff $E(frac {xi} {tau} I_A)=0$ for any set $A$ of the type ${tau in E}$ where $E$ is a Borel set in $mathbb R$. It is enough
          to prove this when $E$ is an open interval and in this case what we need is exactly the hypothesis.



          PS: there is no guarantee that $frac {xi} {tau}$ is integrable. To avoid this problem we can replace $xi$ by $xi I_{{tau >epsilon}}$ in the proof and let $epsilonto 0$ at the end.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Thank you very much for your quick reply. Isn't it necessary for $E(ximidtau)=0iff Eleft(fracxitaumidtauright)=0$ to hold that we have $Eleft|fracxitauright|<infty$ which does not have to be the case here.
            $endgroup$
            – Agnetha Timara
            Dec 27 '18 at 10:29










          • $begingroup$
            @AgnethaTimara You are right but that can easily be circumvented. It is enough to to show that $E( xi I_{tau >epsilon}|tau)=0$ for each $epsilon >0$.
            $endgroup$
            – Kavi Rama Murthy
            Dec 27 '18 at 10:37












          • $begingroup$
            Thanks again for clarifying that. Can you also tell me why it suffices to show that for a generating set of the $sigma$ algebra $sigma(tau)$. Is there a theorem I can refer to or can I use dominated convergence?
            $endgroup$
            – Agnetha Timara
            Dec 27 '18 at 11:44










          • $begingroup$
            @AgnethaTimara The $pi -lambda$ Theorem or Dynkin's Theorem is ideally suited for this kind of situation. It tells you immediately why it is enough to consider intervals.
            $endgroup$
            – Kavi Rama Murthy
            Dec 27 '18 at 11:47










          • $begingroup$
            Sorry for the many questions. I really appreciate your help. These theorems tell me that the intervals generate the Borel $sigma$ algebra. I don't understand yet why $int_{{x<tauleq y}} Z=0$ implies $int_{{tauin E}} Z=0$ for $Ein mathcal B$ and $Z$ a random variable. Obviously $E$ can be represented as a union / intersection of intervals but for example $int_A Z=0$ for $Ain{A_1,ldots A_n}$ does not imply $int_{bigcup_i A_i}Z=0$
            $endgroup$
            – Agnetha Timara
            Dec 27 '18 at 12:16
















          1












          $begingroup$

          The result is true for any positive random variable $tau$. Exponential distribution is note required. $E(xi |tau)=0$ iff $E(frac {xi} {tau} |tau)=0$ iff $E(frac {xi} {tau} I_A)=0$ for any set $A$ of the type ${tau in E}$ where $E$ is a Borel set in $mathbb R$. It is enough
          to prove this when $E$ is an open interval and in this case what we need is exactly the hypothesis.



          PS: there is no guarantee that $frac {xi} {tau}$ is integrable. To avoid this problem we can replace $xi$ by $xi I_{{tau >epsilon}}$ in the proof and let $epsilonto 0$ at the end.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Thank you very much for your quick reply. Isn't it necessary for $E(ximidtau)=0iff Eleft(fracxitaumidtauright)=0$ to hold that we have $Eleft|fracxitauright|<infty$ which does not have to be the case here.
            $endgroup$
            – Agnetha Timara
            Dec 27 '18 at 10:29










          • $begingroup$
            @AgnethaTimara You are right but that can easily be circumvented. It is enough to to show that $E( xi I_{tau >epsilon}|tau)=0$ for each $epsilon >0$.
            $endgroup$
            – Kavi Rama Murthy
            Dec 27 '18 at 10:37












          • $begingroup$
            Thanks again for clarifying that. Can you also tell me why it suffices to show that for a generating set of the $sigma$ algebra $sigma(tau)$. Is there a theorem I can refer to or can I use dominated convergence?
            $endgroup$
            – Agnetha Timara
            Dec 27 '18 at 11:44










          • $begingroup$
            @AgnethaTimara The $pi -lambda$ Theorem or Dynkin's Theorem is ideally suited for this kind of situation. It tells you immediately why it is enough to consider intervals.
            $endgroup$
            – Kavi Rama Murthy
            Dec 27 '18 at 11:47










          • $begingroup$
            Sorry for the many questions. I really appreciate your help. These theorems tell me that the intervals generate the Borel $sigma$ algebra. I don't understand yet why $int_{{x<tauleq y}} Z=0$ implies $int_{{tauin E}} Z=0$ for $Ein mathcal B$ and $Z$ a random variable. Obviously $E$ can be represented as a union / intersection of intervals but for example $int_A Z=0$ for $Ain{A_1,ldots A_n}$ does not imply $int_{bigcup_i A_i}Z=0$
            $endgroup$
            – Agnetha Timara
            Dec 27 '18 at 12:16














          1












          1








          1





          $begingroup$

          The result is true for any positive random variable $tau$. Exponential distribution is note required. $E(xi |tau)=0$ iff $E(frac {xi} {tau} |tau)=0$ iff $E(frac {xi} {tau} I_A)=0$ for any set $A$ of the type ${tau in E}$ where $E$ is a Borel set in $mathbb R$. It is enough
          to prove this when $E$ is an open interval and in this case what we need is exactly the hypothesis.



          PS: there is no guarantee that $frac {xi} {tau}$ is integrable. To avoid this problem we can replace $xi$ by $xi I_{{tau >epsilon}}$ in the proof and let $epsilonto 0$ at the end.






          share|cite|improve this answer











          $endgroup$



          The result is true for any positive random variable $tau$. Exponential distribution is note required. $E(xi |tau)=0$ iff $E(frac {xi} {tau} |tau)=0$ iff $E(frac {xi} {tau} I_A)=0$ for any set $A$ of the type ${tau in E}$ where $E$ is a Borel set in $mathbb R$. It is enough
          to prove this when $E$ is an open interval and in this case what we need is exactly the hypothesis.



          PS: there is no guarantee that $frac {xi} {tau}$ is integrable. To avoid this problem we can replace $xi$ by $xi I_{{tau >epsilon}}$ in the proof and let $epsilonto 0$ at the end.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Dec 27 '18 at 11:52

























          answered Dec 27 '18 at 10:11









          Kavi Rama MurthyKavi Rama Murthy

          74.6k53270




          74.6k53270












          • $begingroup$
            Thank you very much for your quick reply. Isn't it necessary for $E(ximidtau)=0iff Eleft(fracxitaumidtauright)=0$ to hold that we have $Eleft|fracxitauright|<infty$ which does not have to be the case here.
            $endgroup$
            – Agnetha Timara
            Dec 27 '18 at 10:29










          • $begingroup$
            @AgnethaTimara You are right but that can easily be circumvented. It is enough to to show that $E( xi I_{tau >epsilon}|tau)=0$ for each $epsilon >0$.
            $endgroup$
            – Kavi Rama Murthy
            Dec 27 '18 at 10:37












          • $begingroup$
            Thanks again for clarifying that. Can you also tell me why it suffices to show that for a generating set of the $sigma$ algebra $sigma(tau)$. Is there a theorem I can refer to or can I use dominated convergence?
            $endgroup$
            – Agnetha Timara
            Dec 27 '18 at 11:44










          • $begingroup$
            @AgnethaTimara The $pi -lambda$ Theorem or Dynkin's Theorem is ideally suited for this kind of situation. It tells you immediately why it is enough to consider intervals.
            $endgroup$
            – Kavi Rama Murthy
            Dec 27 '18 at 11:47










          • $begingroup$
            Sorry for the many questions. I really appreciate your help. These theorems tell me that the intervals generate the Borel $sigma$ algebra. I don't understand yet why $int_{{x<tauleq y}} Z=0$ implies $int_{{tauin E}} Z=0$ for $Ein mathcal B$ and $Z$ a random variable. Obviously $E$ can be represented as a union / intersection of intervals but for example $int_A Z=0$ for $Ain{A_1,ldots A_n}$ does not imply $int_{bigcup_i A_i}Z=0$
            $endgroup$
            – Agnetha Timara
            Dec 27 '18 at 12:16


















          • $begingroup$
            Thank you very much for your quick reply. Isn't it necessary for $E(ximidtau)=0iff Eleft(fracxitaumidtauright)=0$ to hold that we have $Eleft|fracxitauright|<infty$ which does not have to be the case here.
            $endgroup$
            – Agnetha Timara
            Dec 27 '18 at 10:29










          • $begingroup$
            @AgnethaTimara You are right but that can easily be circumvented. It is enough to to show that $E( xi I_{tau >epsilon}|tau)=0$ for each $epsilon >0$.
            $endgroup$
            – Kavi Rama Murthy
            Dec 27 '18 at 10:37












          • $begingroup$
            Thanks again for clarifying that. Can you also tell me why it suffices to show that for a generating set of the $sigma$ algebra $sigma(tau)$. Is there a theorem I can refer to or can I use dominated convergence?
            $endgroup$
            – Agnetha Timara
            Dec 27 '18 at 11:44










          • $begingroup$
            @AgnethaTimara The $pi -lambda$ Theorem or Dynkin's Theorem is ideally suited for this kind of situation. It tells you immediately why it is enough to consider intervals.
            $endgroup$
            – Kavi Rama Murthy
            Dec 27 '18 at 11:47










          • $begingroup$
            Sorry for the many questions. I really appreciate your help. These theorems tell me that the intervals generate the Borel $sigma$ algebra. I don't understand yet why $int_{{x<tauleq y}} Z=0$ implies $int_{{tauin E}} Z=0$ for $Ein mathcal B$ and $Z$ a random variable. Obviously $E$ can be represented as a union / intersection of intervals but for example $int_A Z=0$ for $Ain{A_1,ldots A_n}$ does not imply $int_{bigcup_i A_i}Z=0$
            $endgroup$
            – Agnetha Timara
            Dec 27 '18 at 12:16
















          $begingroup$
          Thank you very much for your quick reply. Isn't it necessary for $E(ximidtau)=0iff Eleft(fracxitaumidtauright)=0$ to hold that we have $Eleft|fracxitauright|<infty$ which does not have to be the case here.
          $endgroup$
          – Agnetha Timara
          Dec 27 '18 at 10:29




          $begingroup$
          Thank you very much for your quick reply. Isn't it necessary for $E(ximidtau)=0iff Eleft(fracxitaumidtauright)=0$ to hold that we have $Eleft|fracxitauright|<infty$ which does not have to be the case here.
          $endgroup$
          – Agnetha Timara
          Dec 27 '18 at 10:29












          $begingroup$
          @AgnethaTimara You are right but that can easily be circumvented. It is enough to to show that $E( xi I_{tau >epsilon}|tau)=0$ for each $epsilon >0$.
          $endgroup$
          – Kavi Rama Murthy
          Dec 27 '18 at 10:37






          $begingroup$
          @AgnethaTimara You are right but that can easily be circumvented. It is enough to to show that $E( xi I_{tau >epsilon}|tau)=0$ for each $epsilon >0$.
          $endgroup$
          – Kavi Rama Murthy
          Dec 27 '18 at 10:37














          $begingroup$
          Thanks again for clarifying that. Can you also tell me why it suffices to show that for a generating set of the $sigma$ algebra $sigma(tau)$. Is there a theorem I can refer to or can I use dominated convergence?
          $endgroup$
          – Agnetha Timara
          Dec 27 '18 at 11:44




          $begingroup$
          Thanks again for clarifying that. Can you also tell me why it suffices to show that for a generating set of the $sigma$ algebra $sigma(tau)$. Is there a theorem I can refer to or can I use dominated convergence?
          $endgroup$
          – Agnetha Timara
          Dec 27 '18 at 11:44












          $begingroup$
          @AgnethaTimara The $pi -lambda$ Theorem or Dynkin's Theorem is ideally suited for this kind of situation. It tells you immediately why it is enough to consider intervals.
          $endgroup$
          – Kavi Rama Murthy
          Dec 27 '18 at 11:47




          $begingroup$
          @AgnethaTimara The $pi -lambda$ Theorem or Dynkin's Theorem is ideally suited for this kind of situation. It tells you immediately why it is enough to consider intervals.
          $endgroup$
          – Kavi Rama Murthy
          Dec 27 '18 at 11:47












          $begingroup$
          Sorry for the many questions. I really appreciate your help. These theorems tell me that the intervals generate the Borel $sigma$ algebra. I don't understand yet why $int_{{x<tauleq y}} Z=0$ implies $int_{{tauin E}} Z=0$ for $Ein mathcal B$ and $Z$ a random variable. Obviously $E$ can be represented as a union / intersection of intervals but for example $int_A Z=0$ for $Ain{A_1,ldots A_n}$ does not imply $int_{bigcup_i A_i}Z=0$
          $endgroup$
          – Agnetha Timara
          Dec 27 '18 at 12:16




          $begingroup$
          Sorry for the many questions. I really appreciate your help. These theorems tell me that the intervals generate the Borel $sigma$ algebra. I don't understand yet why $int_{{x<tauleq y}} Z=0$ implies $int_{{tauin E}} Z=0$ for $Ein mathcal B$ and $Z$ a random variable. Obviously $E$ can be represented as a union / intersection of intervals but for example $int_A Z=0$ for $Ain{A_1,ldots A_n}$ does not imply $int_{bigcup_i A_i}Z=0$
          $endgroup$
          – Agnetha Timara
          Dec 27 '18 at 12:16


















          draft saved

          draft discarded




















































          Thanks for contributing an answer to Mathematics Stack Exchange!


          • Please be sure to answer the question. Provide details and share your research!

          But avoid …



          • Asking for help, clarification, or responding to other answers.

          • Making statements based on opinion; back them up with references or personal experience.


          Use MathJax to format equations. MathJax reference.


          To learn more, see our tips on writing great answers.




          draft saved


          draft discarded














          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3053763%2fconditional-expectation-product-of-random-variables%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown





















































          Required, but never shown














          Required, but never shown












          Required, but never shown







          Required, but never shown

































          Required, but never shown














          Required, but never shown












          Required, but never shown







          Required, but never shown







          -

          Popular posts from this blog

          Bundesstraße 106

          Image
          Dieser Artikel beschreibt die Bundesstraße 106 in Deutschland. Zur gleichnamigen Straße in Österreich siehe Mölltal Straße. Vorlage:Infobox hochrangige Straße/Wartung/DE-B Bundesstraße 106 in Deutschland Karte Basisdaten Betreiber: Deutschland  Bundesrepublik Deutschland Straßenbeginn: Wismar ( 53° 54′  N , 11° 24′  O 53.900536 11.396875 ) Straßenende: Schwerin ( 53° 36′  N , 11° 24′  O 53.592785 11.404892 ) Gesamtlänge: 40,7 km Bundesland: Mecklenburg-Vorpommern Ausbauzustand: zweistreifig [1] Bundesstraße 106 in Schwerin (Umgehungsstraße) Straßenverlauf Land Mecklenburg-Vorpommern Landkreis Nordwestmecklenburg Ortsumgehung Wismar Wallensteingraben über   Dorf Mecklenburg Wallensteingraben Groß Stieten Bad Kleinen OT Niendorf Bahnstrecke Lübeck–Bad Kleinen Zickhusen Lübstorf Klein Trebbow OT Kirch Stück ...
          Read more

          Verónica Boquete

          Image
          Verónica Boquete im Trikot der Nationalmannschaft (2013) Personalia Name Verónica Boquete Giadans Geburtstag 9. April 1987 Geburtsort Santiago de Compostela, Spanien Größe 159 cm Position Mittelfeld / Sturm Juniorinnen Jahre Station 2001–2005 Xuventú Aguiño Frauen Jahre Station Spiele (Tore) 1 2005–2008 Prainsa Zaragoza 83 (40) 2008–2010 Espanyol Barcelona 53 (30) 2010 Buffalo Flash 9 0 (9) 2010 Chicago Red Stars 3 0 (1) 2010–2011 Espanyol Barcelona 26 (41) 2011 Philadelphia Independence 13 0 (5) 2011 Energija Woronesch 5 0 (1) 2012–2014 Tyresö FF 39 (14) 2014 Portland Thorns FC 15 0 (4) 2014–2015 1. FFC Frankfurt 23 0 (7) 2015–2016 FC Bayern München 9 0 (0) 2016–2018 Paris Saint-Germain 31 0 (8) 2018 Peking BG Phoenix 2019– Utah Royals FC 0 0 (0) Nationalmannschaft Jahre Auswahl ...
          Read more

          Ida-Boy-Ed-Garten

          Image
          Die Lage des Ida-Boy-Ed-Gartens, rot markiert Der Ida-Boy-Ed-Garten Der Ida-Boy-Ed-Garten, Blick von der Wakenitzmauer durch das Tor in der Stadtmauer Der Ida-Boy-Ed-Garten ist eine Straße der Lübecker Altstadt. Inhaltsverzeichnis 1 Lage 2 Geschichte 3 Literatur 4 Weblinks Lage | Der etwa 90 Meter lange Ida-Boy-Ed-Garten befindet sich an der nördlichen Spitze der Altstadtinsel, vor dem Burgtor. Die in einem Bogen durch eine Grünanlage hangaufwärts führende Straße verbindet die Wakenitzmauer mit der Burgtorbrücke. Geschichte | Der heutige Ida-Boy-Ed-Garten wurde ursprünglich als Verlängerung der Wakenitzmauer angelegt. 1903 wurde in den erhaltenen Teil der mittelalterlichen Stadtmauer, an dem die Wakenitzmauer bei der Einmündung der Kaiserstraße begann, ein Tor mit einem großen Bogen für die Durchführung der Straße und zwei kleineren seitlichen Öffnungen für Fußgänger gebrochen, um für die neu errichteten Wohnhäuser in diesem Bereich ...
          Read more